The Geometry And Topology Of Three-Manifolds

Citation preview

The Geometry and Topology of Three-Manifolds

William P. Thurston

1979

The Geometry and Topology of 3-manifolds William P. Thurston

Introduction

These notes (through p. 9.80) are based on my course at Princeton in

79.

Large portions were written by Bill Floyd and Steve Kerckhoff.

1978-

Chapter 7,

by John Milnor, is based on a lecture he gave in my course; the ghostwriter

was Steve Kerckhoff. next academic year.

The notes are projected to continue at least through the

The intent is to describe the very strong connection between

geometry and lowadimensional topology in a way which will be useful and accessible (with some effort) to graduate students and mathematicians working

in related fields, particularly

3-manifolds and Kleinian groups.

Much of the material or technique is new, and more of it was new to me.

As a consequence, I did not always know where I was going, and the discussion often tends to wander. The countryside is scenic, however, and it is fun to

tramp around if you keep your eyes alert so you don't get lost. The tendency

to meander rather than to follow the quickest linear route is especially pronounced in chapters

8

and

9,

where I only gradually saw the usefulness of "train

tracks" and the value of mapping out scum global information about the

structure

of the set of simple geodesics on surfaces.

I would be grateful to hear any

suggestions or corrections from readers,

since changes are fairly easy to make at this stage.

In particular, bibliographical

information is missing in many places, and I would like to solicit references

(perhaps in the form of preprints) and historical infommation.

Table of Contents

- 1.8 pp. 2.1 - 2.22

1. Some topological constructions for 3-manifolds

2. Geometry

-

elliptic, parabolic and hyperbolic

2.1 The

..............

2.5 2.6

Upper half space

2.h The

projective, or Klein model

2.5 The

sphere of imaginary radius

2.6

1.1

.................... 2.3

Poincare’ disk

2.2 The Southern hemisphere

2.3

pp.

Trigonometry

2.7

............................. 2.10

............,......................

3. Geometric Structures on

manifolds

2.12 pp.

3.1

- 3.2M

3.1 A

hyperbolic structure on the figure eight knot complement .

3.6

3.2 A

hyperbolic manifold with geodesic boundary

3.7

3.3 The Whitehead

link complement

3.9

3 h The Borromean

rings complement

3.11

3.5 The

developing map

......................................... 3.12

3.6

A sufficient condition for completeness

3.7

Conditions for completeness

3.8

Hbrospheres

3.9

Hyperbolic surfaces obtained from ideal triangles. The

.......

3.17

............................................... 3.18

completion

3.10

.................... 3.13

H

3.20

.....

Hyperbolic manifolds obtained by glueing ideal polyhedra ..

h. Hyperbolic Dehn surgery

.........................

H3

pp. h.l

3.22

- h.56

h.1

Shapes of ideal tetrahedra in

h.2

Glueing consistency conditions

h.3

Hyperbolic structures on the figure eight knot complement .. h.8

h.l

h.5

h.h The completion of hyperbolic 3-manifolds obtained from

h.l3

ideal polyhedra

h.5

h.6 Dehn surgery on

h.7

............... h.16

The generalized Dehn surgery invariant

Theorem.

(S3

the figure eight knot

- £3 )(u,k)

.... ................... h.18

has a hyperbolic sturucture unless

(in , 1x) = (1 , o) , (o , 1) (1 , 1) , (2 , 1) , (3 , 1) or

(M , l) ............................................ h.23

h.8 Degeneration of hyperbolic structures ..........

h.9 h.lo

Incompressible surfaces in

(S3

-

E: )(u X)

S3 - I:

3

(in , :A) = (o , 1) or

......

is irreducible and Haken only when

(h , 1)

............................ h.h1

Flexibility and rigidity of geometric structures ................... pp.

5.1

Nearby structures are determined by holonomy

5.2 A

crude dimension count

5.5 The

5.6

5.1

- 5.6h

........................5.2

.............................................5.3

........................................... 5.6 Special algebraic properties of groups of isometries of H3 ....... 5.1M

5.3 Teichmfiller

5.h

h.29 h.38

Theorem characterizing hyperbolic foliations of closed 3-manifolds.

4.11 Theorem.

5.

h.23

.

space

.......

.... 5.22

dimension of the deformation space of a hyperbolic manifold

Theorem

....

counting dimension for deformations of hyperbolic

3-manifolds .........................................

5.7 Mostow's

..... 5.22

theorem. (rigidity of hyperbolic manifohis with finite

volume)... ........... ........................;................

5.29

0.1!-

5.8

Generalized Dehn surgery and hyperbolic structures .............. 5.32

5.9 AProof ofMostow's

5.10 A decomposition of hyperbolic manifolds . 5.11 Complete hyperbolic manifolds with bomded 5.12 J¢rgensen's 6. Gromov's

5.56 5.61

volume

.....

............

invariant

Groinov's theorem

6.3

Gromov's proof of Mostow's theorem

6.’+

Strict version of Gromov's theor-

6.5

Manifolds with boundary

6.6

ordinals

6.7

Comensurability..... ...........

6.8 Some

examples

Computation of volume

7.1 The

6.1

- 6.118

....................... 6.1

.................................... 6.7

6.2

7.2

.......... 5.51

......

invariant and the volume of a hyperbolic manifold ....... pp.

6.1 Gromov's

7.

theorem

5.39

.........

theorem .......

.......

.....................

...... 6.11.

.,.......... 6.12

......................................... 6.19 '

6.25

..........

.............

.......

......

6.28

.....

................................................... 6.33 by J. W. Milnor

Lobachevsky function

..................... pp. 7.1

J( (9)

..................

7.1

7.6

.

.

Volumes of some polyhedra.

7.3 Some manifolds ......................................

- 7.21

......

7.13

.................................. ..... 7.17 References for chapter 7 ... .- ........................ .......,........ 7.21

7.1+

8

.

Arithmetic examples

Kleinian groups............... .......

pp.

8.1-8.77

................................................... 8.1

8.1 The

limit set

8.2 The

domain of discontinuity

8.3

........

Convex hyperbolic manifolds

.....................

............. 8.5

..................................... 8.9

8.1+

Geometrically finite groups

8.5 The geometry

.....

8.15 8.22

of the bomdary of the convex hull

8.6

Measuring laminations

8.29

8.7

Quasi-Fuchsian groups ....................

8.33

8.8

Uhcrumpled surfaces............ ......

..................

8.9 The structure

8.10

of geodesic laminations:

Realizing laminations in

8.1.1 The structure

8 .12 Eamonic 9.

................

Algebraic

of cusps

.....

9.1 Limits

8.51

train tracks

8.57 8.69 8.73

3-manifolds ....

-. ..........

functions and ergodicity

convergence ............: ........................... pp. 9.1

-?

of discrete groups ,....................

9.2 Theorem

9.1

- strong convergence of sequences of quasi-Fuchsian 9-9

groups...... .....

9.3 The

8.1m-

ending of an end

9.15

.

9.11-

Taming the topology of an end

9.25

9.5

Interpolating negatively curved surfaces .......................

9.28

9.6

Strong convergence from algebraic convergence ..................

9.52

9.7 Realization

of geodesic laminations for surface groups with

extra cusps, with a digression on stereographic

.......................

coordinates .........................

9.58

and

The theme I intend to develop is that topology and geometry, in dimensions

up through 3, are very intricately related.

Because of this relation, many

questions which seem utterly hopeless from a purely

can be fruitfully studied. eventually

topological

point of view

It is not totally unreasonable to hope that

all 3-manifolds will be understood in a systematic way.

In any

case, the theory of geometry in 3-manifolds promises to be very rich, bringing together many threads.

51

Before discussing geometry, 1 will indicate some topological constructions

yielding diverse 3-manifolds, which appear to be very tangled.

0. Start with the three sphere

S3 ,

which may be easily visualized as

together with one point at infinity.

1. Any knot ( = a closed simple curve) or link ( = a union of disjoint closed simple curves) may be removed. These examples can be made compact by removing the interior of a

tubular neighborhood of the knot or link.

113

1.2. The complement of a knot can be very enigmatic, if you try to think about it from an intrinsic point of view. Papakyriakopoulos proved that a knot complement has fundamental group

intuitively clear, but

Z iff the knot is trivial. This may seem

Justification

for this intuition is difficult.

It is

not known whether knots with homeomorphic complements are the same.

2. Cut out atubular neighborhood of a knot or link, and glue it backin by a different identification.

This is called Q§hn_surge£y.

ways to do this, because the torus has many diffeomorphisms.

of the kernel of the inclusion map

Tr1 (T2) +

n1

resulting 3—manifold determines the 3-manifold.

There are

many

The generator

(solid torus)

in the

The diffeomorphism can be

chosen to make this generator an arbitrary primitive ( = indivisible non-zero)

element of Z 6 Z . It is well defined up to change in sign.

Every oriented 3—manifold can be obtained by this construction, ( cf.

[Lickorish,

, ] for instance.) It is difficult, in general, to tell

much about the 3-manifold resulting from this is it simply connected?

construction.

When is it irreducible?

When, for instance,

(MEmeans every

embedded two sphere bounds a ball).

Note that the homology of the three-manifold is a very insensitive invariant. The homology of a knot complement is the same as the homology of a circle, so when Dehn surgery is performed, the resulting manifold always has a cyclic first homology group.

that

If generators for

Z) 9

Z =

#1 (T2)

(l , 0) generates the homology of the complement and

then any Dehn surgery with invariant

83 —

so‘

(O , l) is trivial

(l , n) yields a homology sphere.

3. Branched coverings. If L is a covering space of

are chosen

link, then any finite-sheeted

L can be compactified in a canonical way be adding

circles which cover covering of

L to give a closed manifold, M . 'M is called a branched

S3 over L . There is a canonical projection

which is a local diffeomorphism away simplest branched coverings of

S3

p-1

from

p : M +

3

K arrows

glue the faces of one tetrahedron to the other so

For instance, A is matched with A'. All the

are identified and all the

resulting complex has

—f+——9

arrows are identified, so the

2 tetrahedra, h triangles, 2 edges and l vertex.

Its Euler characteristic is +l , and (it follows that) a neighborhood of the vertex is the cone on a torus. Let M be the manifold obtained by removing the vertex.

It

turns out that

this manifold is homeomorphic with the complement of a

figure eight knot.

"Figure

eight

knot."

1.6

"Another view of the figure eight

This knot is

familiar

knot."

from extension cords, as the most commonly occurring knot,

after the trefoil knot

In order to see this homeomorphism we can draw a more suggestive picture of the figure eight knot,

4g

"Tetrahedron with

figure eight knot,

viewed from above."

arranged along the l-skeleton of a tetrahedron.

The knot can be spanned by a

2-complex, with two edges, shown as arrows, and

h 2-cells, one for each face

of the tetrahedron, in a more

- or - less obvious way: 3

I

This

picture illustrates

the typical way in which a

2—cell is attached.

Keeping in mind that the knot is not there, the cells are triangles with deleted

vertices.

The two complementary regions of the two-complex are the tetrahedra,

with deleted vertices.

We will return to this example later. For now, it serves to illustrate the need for a systematic way to compare and to recognize manifolds.

3933p

Suggestive pictures can also be deceptive. A trefoil knot can similarly

be arranged along the

l-skeleton of a tetrahedron

then 1.8.

From the picture, a cell-division of the complement is produced. however, the

3-cells are not tetrahedra.

The boundary of a 3-cell, flattened out on the plane.

In this case,

52.

There are three kinds of geometry which possess a notion of distance,

and which look the same from any viewpoint with your head turned in an orientation: these are elliptic geometry (or spherical geometry),

Euclidean or

parabolic geometry, and hyperbolic or Lobachevskiian geometry.

The underlying spaces of these three geometries are naturally Riemannian manifolds of constant sectional curvature resp. +1 , 0 , and -l . Elliptic

n-space is the n-sphere, with antipodal points identified.

TOpologically it is projective sphere.

n-space, with geometry inherited from the

The geometry of elliptic space is nicer than that of the sphere

because of the elimination of the pairs of identical, antipodal figures which always pop up in spherical geometry.

Thus, spy.two points in

elliptic space determines a unique line, for instance. object moving away from you

a distance of

2"

appears

smaller and smaller, until it reaches

Then, it starts looking larger and larger and optically,

it is in focus behind you. it

appears

In the sphere, an

Finally, when it reaches a distance of

W

so large that it wOuld seem to surround you entirely.

,

In elliptic space, on the other hand, the maximum distance is that apparent size is a monotone decreasing

g-

, so

function of distance. It

would none—the—less be distressing to live in elliptic space, since you would always be confronted with an image of yourself, turned inside out, upside down and filling out the entire background in your field of view. Euclidean space is familiar to all of us, since it very closely approximates

the geometry of the space in which we live, up to moderate distances. Hyperbolic space is the least familiar to most people.

revolution in

Certain surfaces of

IR 3 have constant curvature -l and so give an idea of

the local picture of the hyperbolic plane.

IE

_.

mz EG e i s

The simplest of these is the pseudosphere, the surface of revolution generated by a tractrix.

A tractrix is the track of a box of stones which

(O , 1) and is dragged by a team of oxen walking along the

starts at

x-axis and pulling the box by a chain of unit length.

Equivalently,

this curve is determined up to translation by the property that its

tangent lines meet the

x axis a unit distance from the point of tangency.

The pseudosphere is not complete, however

-

it has an edge, beyond which it

cannot be extended. Hilbert proved the remarkable theorem that pg complete

C2‘

surface

with

curvature -1 can exist in

I13 .

In spite of this,

convincing physical models can be constructed.

We must therefore resort to distorted pictures of hyperbolic space.

Just as it is convenient to have different maps of the earth for understanding various aspects of its geometry:

for seeing shapes,for

comparing areas, for plotting geodesics in navigation; so it is useful

to have several maps of hyperbolic space at our disposal.

2.1 The Poincaré disk model.

Let

Dn

n of D

denote the disk of unit radius in Euclidean

can be taken as a map of hyperbolic space

Hn

n—space.

The interior

. A hyperbolic

line in the model is any Euclidean circle which is orthogonal to

a hyperbolic

2-plane is a Euclidean sphere orthogonal to

BDn;

BDn

;

etc.

The words "circle" or "sphere" are here used in the extended sense, to include the limiting case of a line or plane.

This model is conformally

correct, that is, hyperbolic angles agree with Euclidean angles, but distances are greatly distorted.

Hyperbolic arc length

1

ds2

is given

2.h

length and r

2

ds2

by the formula

dx2

, where

is distance from the origin.

162;;

is Euclidean arc

Thus, the Euclidean image of

a hyperbolic object, as it moves away from the origin,shrinks in size

BDn

roughly in proportion to the Euclidean distance from

(when this n 8D , if it

distance is small). The object never actually arrives at

moves with a bounded hyperbolic velocity.

Lines

The sphere

BDn

People is called the sphere at inrini

. It is not actually

in hyperbolic space, but it can be given an interpretation purely in terms of hyperbolic geometry, as follows.

Choose any base point p0 in

Consider any geodesic ray R , as seen from

of a great circle in the visual sphere at

a

pO . R traces out

p0

(since

p0

and R

En

.

a segment determine

2-plane).Thisvisual segment converges to a point in the visual sphere.

If we translate

Hn

so that

pO

is at the origin in the Poincaré disk

model, we see that the points in the visual sphere correspond precisely

to

points

on the sphere at infinity, and that the end of a ray in this

Visual sphere corresponds to its Euclidean endpoint in the

Poincare

disk

model.

2.2 The southern hemisphere. The Poincaré disk as a hyperbolic

Dnc IR11

is contained in the Poincaré disk

n + 1 space.

n-plane in hyperbolic

Stereographic

projection (Euclidean)

sends the Poincaré disk

Dn

Dn+lC IRn+l

from the north pole of

to the southern hemisphere of

+ 1

BDn

Dn+1

[Rn+5.

mil Soufkaf“

Hemlsfihé’rfe

Thus hyperbolic lines in

to the equator

Sn _

l

the Poincaré

disk go to circles on

S? orthogonal

.

There is a more natural construction for this map, using only + C: l , consider the hyperbolic geometry. For each point p in

Hn

hyperbolic ray perpendicular to

En

at

En

p , and downward normal.

This

,

2.6

"l

ray converges to a point on the sphere at infinity, which is the same as

the Euclidean stereographic image of p.

\ n. be

\W

W

2.3. The upper half space model. This is closely related to the previous two, but it is often more convenient

for computation or for constructing pictures. sphere

space

-J

of

Sn

Sn

in

xn 1 0

IRn+l

so that the southern hemisphere lies in the half-

IR n+1 . Now stereographic projection from the top

(which is now on the equator) sends the southern hemisphere to the

upper half space LJ

in

To obtain it, rotate the

xn > 0

in

IRn+l_

U .

,

(J

(

2.7

(I T

”If

i

wM

. U .,

a;2.

\I

.‘l’ 11%,,

.n.,

C

.

\

(3*

4.5!

A hyperbolic line, in the upper half-sspace, is the bounding plane

1811-1C IR:1 .

a

circle perpendicular to

The hyperbolic metric is

(15:2-an dx2

Thus, the Euclidean image of a hyperbolic object moving toward size precisely proportional to the Euclidean distance from

2.1+ The

annl

has

Tin-l

projective model.

This is obtained by Euclidean orthogonal projection of the southern hemisphere of

line segments.

Sn

back to

the disk Dn

. Hyperbolic lines become Euclidean

This model is useful for understanding incidence in a

configuration of lines and planes.

Unlike the previous three models, it

fails to be conformal, so that angles and shapes are distorted.

It is better to regard this projective model to be contained not in

2.8

Euclidean space, but in projective space.

The projective model is very

natural from a point of view inside hyperbolic picture of a hyperplane, hovering above

Hn

in

En

Hn+l

n + 1 space: it gives a

, in true perspective. Thus, an observer ,

looking down, sees

En

as the interior of

a disk in his visual sphere. As he moves farther up, this visual disk shrinks, as he moves down, it expands, but (unlike in Euclidean space)

the visual radius of this disk is always strictly less than

line on

En

r/2 . A

appears visually straight. It is possible to give an

intrinsic meaning within hyperbolic geometry for the points outside the sphere at infinity in the projective model.

For instance, in the two-

dimensional projective model, any two lines meet somewhere. The conventional

sense of meeting means to meet inside the sphere at infinity (at a tinit e

point). If the two lines converge in they meet on the circle at

infinity,

Otherwise, the two lines are called

common perpendicular

the

visual circle, this means that

and they are called parallels .

we;

they have a unique

L and they meet in some point x in the Moebius

band outside the circle at infinity. Any other line perpendicular pg_m£

passes thropgh x , and any line through x is p___a*mndicular 39_ L.

2.8.a.

Evenly

lines

The region inside the circle is a plane, with a base line and a family of its perpendiculars, spaced at a distance of .051 fundamental units, as measured along the base line shown in perspective in hyperbolic 3-space (or in the projective ary meet ing point beyond model). The lines have been extended t0 their the her izon. U , the observer, is directly ab0V8 the X (which is .881 fundamental units HWY 51‘0“ the base 11ne). To see the view from different heights, us the following table:

To see the view of U at a height of

hold the picture a distance of

—

2 units

11"

3 units ’4 units 5 units

27"

10 units 20 units

2,523'

6' 17'

10,528.75

(28 cm.) (69 cm.) (191 cm.) (519.cm.) (771 m.) miles

(16,981 km.)

For instance, you may imagine that the fundamental distance is 10 meters. Then the lines are spaced about like railroad ties. Twenty units is 200

meters: U is in a hot air balloon.

To prove this, consider hyperbolic 2-space as a plane PC

33 .

Construct the plane Q through L perpendicular to P . Let U be an observer

e:

H3. ,Drop a perpendicular

Now if K is any line in by

U and K

P perpendicular to L , the plane determined

is perpendicular to

visual line determined by

M from U to the plane Q .

Q , hence contains M ; hence the

K in the visual sphere of U passes through

the visual point determined by

K . The converse is similar.

2.10

a one—to-one

This gi

e at infinity

the

s

L corre /

.r

S

lerly, there

w’8utside

. “.- 4'

the sphere

The

afl

infinity

a point

L in

int of all its perpendiculars.

s in

orreSpondence between

Sphere

HI1

and hyperplanes

sponds to the union of

p

of imaginary radius.

Euclidean

A sphere in

space with radius r

I

!

space should be

a sphere of 1 f .. we use an indeflnlte . . . . 1nterpretatlon, met

hyperbolic

in

hyperplanes

general) the set

outside

determined by hyperplanes through I

all points

2.5

in

the common intersection

is a

at of points x

between the

.

IRn+l. The sphere of radius .

hyperb0101d

i

,1" 2 + ... + xn2 xn+12 fxl —

I"

constant curvature 12 r

Thus,

i . To give this a reasonable

2

_— dxl2 + ... + dxn2 — dxn+12

the origin in this metric is the

K

\

\\ \

./

IJ

2.11

The metric

dx2

restricted to this hyperboloid is positive definite, and

curvature

it is not hard to check that it has constant

Any plane through the origin is

dx2 —

-l .

orthogonal'to the hyperboloid,

so it follows from elementary Riemannian geometry that it meets the hyperboloid in a geodesic.

The projective model for hyperbolic space is reconstructed

by projection of the hyperboloid from the origin to a hyperplane in Conversely, the quadratic form

from the projective model.

x12

+ ... +

xnz - xn+12

Tin

.

can be reconstructed

To do this, note that there is a unique

quadratic equation of the form

i,j=l

defining the sphere at infinity in the projective model.

this equation gives a quadratic form of type desired. Any isometry of the quadratic form

Homogenization of

IR'n+l , as x12 + ... xn 2 - xn_12 (n , 1) in

induces an isometry of the hyperboloid, and hence any projective transformation

of

IPn

which

preserves

hyperbolic space.

the sphere at infinity induces an isometry of

This contrasts with the situation in Euclidean geometry,

where there are many projective self-homeomorphisms: the affine transformations.

In particular, hyperbolic space has no similarity

transformations except isometries.

This is true also for elliptic space.

This means that there is a well-defined unit of measurement of distances in

hyperbolic geometry. We shall later see how this is related to 3-dimensional

topology, giving a measure of the

"size" of manifolds.

2.12

2.6

Trigonometry.

Sometimes it is convenient to have formulas for hyperbolic geometry, and

not just pictures. For this purpose, it is convenient to work with the description of hyperbolic space as one sheet of the "sphere" of radius

— X1

. with respect to the quadratic form Q(X) _2 + ... +

Iin+l

.

+

film

inner product

x: - Xn+1

i

2.‘ 1n

1 , equipped with this quadratic form and the associated

X

'Y

n

X. Y.

=_

_

X 1 Y 1 , is called

En a l .

i:

First we will describe the geodesics on level sets of

we

Q . Suppose that

Kt

is such a geodesic, with speed

Sr = {X s = V

: Q(X) =

Q(K )

may differentiate the equations

_

Xt Xt to obtain

Kt 'Kt

. _ Xt. Kt -

2

r

Xt Xt

= 0

- Kt Xt --

Xt Xt

I

I

Since any geodesic must lie in a a linear combination of

2.6.1

X

=

Xt

and

2

s

0

2

-5

2-dimensional subspace,

it

, and we

Xt

have

X

This differential equation, together with the initial conditions

must be

r2}

2.13

determines the geodesics. Given two vectors

X and Y

in

En’l

, if X and I have non-zero

length we define the quantity

c (X

IIXII

where

= /F_X_T_X—

, Y)

X

'Y

llxll

°

Irll

is positive real or positive imaginary. Note that

c (X , Y)

= c (1X , uY) , where A and u are positive constants, that

c(-X , Y)

= —c (x , r), and that c (x , x) = 1 . In Euclidean space

En+l

, c(X , I) is the cosine of the angle between X and I

En’l

In

there are several cases.

We identify vectors on the positive sheet of If Y

hyperbolic space.

to the subspace that

Y‘

intersects

the notation

Emsby_ 2.6.2.

Y‘"

If

Y‘

with

Q restricted

(n—l , l) . This

means

and determines a hyperplane. We will use

tat

to denote this hyperplane, piph_phe_normal orien

Y .

x

is any vector of real length, then

is indefinite of type

fin

Si (Xn+1 > 0)

and

ion

(We have seen this correspondence before, in 2.h) .

r

e.an

, then

c(X , r) = cosh d(X , r) where

d(X , Y)

denotes the hyperbolic distance between

To prove this formula, join X to X by a geodesic

and Y .

X

Xt

of unit speed.

2.1h

From 2.6.1 we have

Xt

Xt

so

c c c thus

c(X , xt) =

When

t

If X

XL

2.6.3

-

and Y and

X0

= 0

(xt, xt) - c (xt. It) (x0, x0) - o (x , x0) =

t.

cosh

d(X , Y) , then .L

X

Xt = Y

, giving

2.6.2.

. . are dlstlnct hyperplanes, then

J-

Y'L

intersect

(in En)

Q is positive definite on the subspace

C(X

,Y)2


1 consider an annulus of radii

, then a (G , JRn)

For instance, given a constant l and,

A + e . Identify

neighborhoods of the two boundary components by the map resulting manifold,- topologically, is

IR11

T2

3: -

f Xx. The

Here is another method, due to John Smillie, for constructing affine

T2

structures on

opposite edges of

from any quadrilateral

Q in the plane.

Identify the

Q by the orientation-preserving similarities which carry

one to the other. Since similarities preserve angles, the sum of the angles about the vertex in the resulting complex is

2v , so it has an

T2

affine structure. We shall see later how such structures on

connected with questions

concerning Dehn surgery in

are intimately

3-manifolds.

The literature about affine manifolds is interesting. Milnor [

] showed

that the only closed 2—dimensional affine manifolds are tori and Klein bottles.

The main unsolved question about affine manifolds is whether in

general an affine manifold has Euler characteristic zero.

If G

is the group of isometries of Euclidean space

En

, then a (G , En)-

manifold is called a Euclidean manifold, or often a flat_manifold. Bieberbach proved that a

Euclidean manifold

is finitely covered by a torus.

Furthermore,

a Euclidean structure automatically gives an affine structure, and Bieberbach

proved that closed Euclidean manifolds with the same fundamental group are equivalent as affine manifolds.

If G

is the group 0 (n + l) acting on elliptic space

then we obtain

ellifl ic

Cong ecture eve

I’n

( or on

Sn

) ,

manifolds.

g—manifold

m finite fundamental m g;a elliptic

structure. This conjecture is a stronger version of the Poincaré conjecture; we shall

see the logic shortly. All known 3—manifolds with finite fundamental group certainly have elliptic structures.

3.h As a final example (for the present),when G is the group of isometries of hyperbolic space

En

, then a

(G , fin)-manifold is a

manifold.

'

For instance, any surface of negative Euler characteristic has a hyperbolic structure.

The surface of genus two is an illustrative example.

Topologically, this surface is obtained by identifying the sides of an

octagon, in the pattern above, for instance. An example of a hyperbolic

structure on the surface is obtained from any hyperbolic octagonawhose opposite edges have equal lengths and whose angle sum

in the same pattern. instance.

is

2n , by identifying

There is a regular octagon with angles w/h , for

s/ A regular octagon with angles fi/h , whose sides can be identified to give a surface of genus 2.

3.6 3.1 A_

erbolic structure gg.the figgre eight knot

C

lament

Consider a regular tetrahedron in Euclidean space, inscribed in the unit sphere, so that its vertices are on the sphere. Now interpret this

tetrahedron to lie in the projective model for hyperbolic space, so that it determdnes an ideal hyperbolic simplex: combinatorially, a simplex with its vertices deleted.

60° .

The dihedral angles of the hyperbolic simplex are

This may be seen by extending its faces to the sphere at

they meet in four circles which meet each other in

\

60°

a

, which

angles.

A tetrahedron inscribed in the unit sphere, top view.

By considering the Poincaré disk model, one sees immediately that the angle made by two planes is the same as the angle of their bounding circles on the sphere at infinity.

Take two copies of this ideal simplex, and glue the faces together, in the pattern described in

§1

, using Euclidean isometries, which are also

(in this case) hyperbolic isometries, to identify faces. This gives a hyperbolic structure to the resulting manifold, since the angles add up to

360°

around each edge.

According to Magnus, Hyperbelie Tesselations, this manifold was constructed by

Gieseking

1912

in

R. Riley,

(but without any relation to knots).

, showed that the figure eight knot complement

'

has a hyperbolic structure (which

agrees with

this one).

This manifold also

coincides with one of the hyperbolic manifolds obtained by an arithmetic construction, because the fundamental group of the complement of the figure eight knot is isomorphic to a subgroup of index

where

3.2

m

12 in

PSI.2

( Z Du] ),

is a primitive cube root of unity.

A hmerbolic

manifold with geodesic bouncing

Here is another manifold which is obtained from two tetrahedra. First glue the two tetrahedra along one face; then glue the remaining faces according to this diagram:

|l

\

In the

diagram, one vertex has

been removed so that the polyhedron can be

flattened out in the plane. The resulting complex has only one edge and

one vertex. The manifold M obtained by removing a neighborhood of the vertex is oriented with boundary a surface of genus 2

.

Consider now a one-parameter family of regular tetrahedra in the projective model for

hyperbolic

space, beginning with the

space centered at the origin in Euclidean

tetrahedron

whose vertices are on the sphere at

infinity, and expanding until the edges are all tangent to infinity.

The dihedral angles go from

there is a tetrahedron with along each plane

v‘

300

600 to 00

, so somewhere in between,

dihedral angles.

, where v is a vertex

the sphere at

Truncate this simplex

(outside the unit ball),

to obtain a stunted simplex :

All angles are

900

or

300

with all angles

90° or 30° .

Two copies glued together give a hyperbolic structure for M , where the boundary of M

(which comes from the triangular faces of the stunted

simplices) is totally geodesic.

A_closed h erbolic 3—manifold can be

obtained by doubling this example: i;e; , taking two copies of M

and

glueing them together by the "identity" map on the boundary.

3.3 The Whitehead

link com lament

The Whitehead link may be spanned by a 2—complex which cuts the

complement into

an octahedron, with vertices deleted:

The

1—cells are the three arrows, and the attaching maps for the 2—cells

are indicated by the dotted lines. The 3—ce11 is an octahedron (with

3.10

vertices deleted), and the faces are identified thus:

A

/

(:I A hyperbolic structure may be obtained from a Euclidean regular octahedron inscribed in the unit sphere. Interpreted as lying in the projective model for hyperbolic space, this octahedron is an ideal octahedron with all

dihedral angles

90° .

a—‘Q"

Glueing it in the indicated pattern, again using Euclidean isometries

between the faces (which happen to be hyperbolic isometries as well) gives a hyperbolic

3.h

The

structure for the complement of the Whitehead link.

rings mama

This is spanned by a

2-complex which cuts the complement into two

T\

n

Q

\“n

ideal octahedra:

Borromean rings

A spanning 2—complex

3f

The corresponding glueing pattern two octahedra. Faces are glued to their corresponding faces with 120 rotations, alternating in direction like gears.

3.12

3-5 _Th_e_ developig @Let X be any real analytic manifold, and G a group of real analytic diffeomorphisms of X

. Then an element of G is completely determined X .

by its restriction to any open set of

Suppose that M

(G , X)-manifold. Let

is any

¢i

coordinate charts for M , with maps

Yij

:

¢i (UiO UJ) YiJ

denoted

YiJ ' s

beginning in

U1

where

,

yeG

451

defined on

so they are determined by locally constant maps, also

, of

Ui n

U

G .

into

3

U1

. It is

. Mg,

¢1

of

along a path

a

in M

easy to see, inductively, that on a component of

the analytic continuation of

in M . It of

be

and transition functions

are local G—diffeomorphisms of X

Consider now an analytic continuation

0

,

satisfying

In general the

d

Ui -> X

U1 , U2

4’1

along

cpl SEE.analflicallx

a

is of the form Y o

oi

,

continued alogg avg-z Lag

follows immediately that there is a global analytic

continuation

defined on the universal cover of M . (Use the definition of the

universal cover as a quotient space of the paths on M )- . This map,

D : is called the developing map.

it is an immersion

inducing

the

D

M

+

X

is a local

(G , X)

(G , X)-diffeomorphis'.m

structure on

M

.)

D

is

(i.e.,

3.13 clearly unique up to composition with elements of

G

Although G acts transitively on X in the cases of primary interest, this condition is not necessary for the definition of D . For example, if G

the trivial group and

(G , X)

X is closed then closed

,

the finite-sheeted covers of X

and D

- manifolds are

precisely

is the covering projection.

From this uniqueness property of D , we have in particular that for any covering transformation

gag G

element

D

Since

0

Ta

of

a

a

M

over M , there is some (unique)

such that

Tc TB

g“ o D

0

0

TB

H : c r—+

correspondence

‘ called the is a homomorphism,

ga

o

g8

o D

it

follows that

the

86

holonm

of M .

In general, the holonomy of M need not determine the

(G , X)-structure

on M , but there is an important special case in which it does.

22g. M map.

is a comolete

(G , X)-manifold if D :

(In particular, if X

M +X

is a covering

is simply-connected, this means

D is a

homeomorphism.) If X

is simply connected, then any complete

may easily be reconstructed from the image

as the quotient space

P =

(G , X)-manifold M

H(nl(M))

of the holonomy,

X / P .

Here is a useful sufficient condition for completeness:

3.6

m.Let

G

EEEE Eoup of analytic

diffeomeorphisms acting

transitively o_n_ a manifold X , such that for any xE-X , the isotropy group

Gx

of

x i§_ comEct . Then every closed

Proof. of

Let

(G , X)-manifold M

is complete

Q be any positive definite on the tangent space

Tx(X)

X at some point' x . Average the set of transforms g(Q) , g
0

- ball in the Riemannian metric on

and contractible.

(G , x)- map

M is always convex

D-1 (Be/2 (x) )

is any point in X , then

be a union of homeomorphic copies of

38/2 (x)

in

M

.

must

D evenly covers

X , so it is a covering projection, and M is complete. For example, any closed elliptic 3-manifold has universal cover so any simply-connected elliptic manifold is

S3 .

83 ,

Every closed hyperbolic

manifold or Euclidean manifold has universal cover hyperbolic 3—space

or Euclidean space. Such manifolds are consequently determined by their holonomy.

3.15

The developing map of an affine torus constructed from a quadrilateral (see p. 3.3). The torus is plainly not complete. Exercise: construct other affine toruses with the same holonomy as this one. (Hint: walk once or twice around this page. )

3.16 Even for G and X as in proposition

or not a non-compact

(G , X)

- manifold

3.6

, the question of whether

M is complete can be much more

subtle. For example, consider the thrice-punctured sphere, which is obtained by glueing together two triangles minus vertices in this pattern:

A hyperbolic structure can be obtained by glueing two ideal triangles (with all vertices on the circle at infinity) in this pattern.

Each side of such

a triangle is isometric to the real line, so a glueing map between two sides may be modified by an arbitrary translation; thus, we have a family of

hyperbolic structures in the thrice-punctured sphere parametrized by

IR3 .

(These structures need not be, and are not, all distinct) . Exactly 923_ parameter

zalge_yields a_ Megs h erbolic structure, as we shall see

presently. Meanwhile, we collect some useful conditions for completeness of

structure with

(G , X) as in

3.6.

a (G , X)

For convenience, we fix some natural

-

3.17

metrics on

3.7.

(G , X) -structures.

Proposition.

(G , x) a_s

With

above , then§_ (G , X)-manifold M

complete

c

's

__

Q

(a). M

é)

(b). There

complete as g metric ppace

some a > 0

5

sucp that

e-ball ip M

each closed

i_s

act

Q

(o). For evgy k > 0 , all closed k—ball_s are compact

©

(d)- There.i_s 5 family

M , such that

m.

St+a

contains

Suppose that M

complete.

{St} ';

ta lR ,

g compact sets which

a neighborhood g radius

is metrically complete.

at

in X

St .

Then

M

is also metrically

M

—r

X is a covering map

We will show that the developing map D :

by proving that any path

about

a

ahaust

can be lifted to

M.

In fact, let

Tc [0 , 1] be a maximal connected set for which there is a lifting. Since

D is a local homeomorphism, T is open, and because is closed : hence,

a

can be lifted, so M

this

of

X

xoaX

there is some

s works for all is transitive.

Finally, if M

satisfies

xeX

(b) (c) (6.) => (a) .

such that the ball

since the group

Therefore X

is a complete

a

is metrically complete, T

is complete.

It is an elementary exercise to see that For any point

M

satisfies

G

of

Be(x)

(G , X)

- diffeomorphisms

(a) , (b) , (c) and

(G , X)—manifold, it is covered by

(b) . The proposition follows.

I

is compact;

X

(d)

.

, so it

3.18 3.7.

W-

To analyze what happens near the vertices of an ideal

polyhedron when it is glued together, we need the notion of

(or, in the hyperbolic plane, they-are called horocycles). A horosphere has the limiting shape of a sphere in hyperbolic space, as the radius goes to infinity.

One property which can be used to determine the spheres centered

at a point through

X is the fact that such a sphere is orthogonal to all lines

X . Similarly, if X is a point on the sphere at infinity, the

horo spheres

"centered" at X are

the surfaces orthogonal to all lines through

X . In the Poincaré disk model, a hyperbolic sphere is a Euclidean sphere in the interior of the disk, and a horosphere is a Euclidean sphere tangent

to the unit sphere. The point X of tangency is the center of the horosphere.

Concentric horocycles and orthogonal lines.

3.19

Translation along a line through X Thus,

22$. horo spheres 5;;M.

permutes the horospheres centered at X . The convex region bounded by a horosphere

is a horoball.

For another view of a horosphere, consider the upper half-space model. In this case, hyperbolic lines through the

point

at infinity are Euclidean lines

orthogonal to the plane bounding upper half-space. A horosphere about this point is a horizontal Euclidean plane.

that a horosphere in

fin

From this picture one easily sees

is isometric to Euclidean space

En-1 .

One also

sees that the group of hyperbolic isometries fixing the point at infinity in the upper half-space model acts as the group of Euclidean plane.

. An

isometry

horosphere through X

g

h' . Project

and

22

takes h

. The composition of these two maps is a

xna) dx2 .

centered at

to a concentric

11 and £2

similarity of

h

.

emanating from the point at

Recall that the hyperbolic metric is

This means that the hyperbolic distance between

£1

along a horosphere is inversely proportional to the Euclidean distance

above the bounding plane.

on

any horosphere

h' back to h along the family of parallel lines

infinity in the upper half-space model.

(1 /

h

of hyperbolic space fixing X

Consider two directed lines

ds2 =

of the bounding

One can see this action internally as follows. Let X be

any point at infinity in hyperbolic space, and

X

similarities

21 at heights of

that for any two

hl

The hyperbolic distance between points and

h2

is

[log (h2)

concentric horospheres hl

and

-

h2

log

(hi)

I

X1

and

X2

. It follows

which are a distance

d

3.20

21 and £2 orthogonal to hl and h2 the ratio of the distance between 11 and £2 measured along hl to their

apart, and any pair of lines

distance measured along

h2

is

9i c

exp(d) .

CeJ

Q9.

ha.

/——L/'\

‘"i

Horocycles and lines in the upper half-plane

3.9. Merbolic surfaces obtained fppp.idg§l triangles .

Consider an oriented

S obtained by glueing ideal triangles with all vertices at

surface

infinity, in some

pattern.

Exercise : gll_§pgp; Wiggles are

I

conflent . (Hint:

you can derive this from the fact that a finite triangle is determined by its angles

- see

Let

2.6.8.

Let the vertices pass to infinity, one at a time.)

K be the complex obtained by including the ideal vertices. Associated

with each ideal vertex

v of K , there is an invariant d(v) , defined as

follows. Let h be a horocycle in one of the ideal triangles, centered about a vertex which is glued to v in 8 counter

clockwise

and

about v

"near" this vertex.

Extend h as a horocycle

. It meets each successive

ideal triangle

as a horocycle orthogonal to two of the sides, until finally it re-enters the original triangle as a horocycle

td(v) h'

from h

.

h' concentric with h , at a distance

The sign is chosen to be positive iff the horoball bounded by

in the ideal triangle contains that bounded by

h .

AU)‘0

S i§_

The

iff all

instance, that some invariant

d(v) are 0 .

d(v) < 0 . Continuing h further around v ;

the length of each successive circuit around v factor

0

Y‘OCLiL

A(V,)

ol in

I

Let H'(o.)

77W)

I

d2 generate W) has length

{

I

} is dense in

IR+

H'(c)

I

}

is a cyclic group.

a manifold which is the

and it is obtained by adjoining a circle to

1


(oi , Bi)

happen to be primitive

is the topological manifold

M(ml

’81) ,

with a non—singular hyperbolic structure, so that our extended definition is

If each ratio

compatible with the original.

Pi/qi

in lowest terms, then

M (Plsql),--. '

M

,(Pk’qk) '

(xi/Bi

is the rational number

is topologically the manifold

ThemerOlcs tructure, however , has b l'

, (uk,8k )

1+.l'7

M'—

singularities at the component circles of

2

1r

in

M with cone angles of

+ . . . . (Pi / oi) [Since the holonomy H of the primitive element Pi ai qi bi «1 (Pi) is a pure rotation of angle 2w (pi / mi) 1 .

-

(0.1 , Bi)€Z a

There is also a topological interpretation in case the

In this case, all the cone angles are

although they may not be primitive.

of the form 2 r/

MI

:11 , where

which has branching index

each

:11

Z

:11

is an integer. Any branched cover

around the

of

M'- M

has a

shown on p

h.lO ,

i-th circle of

non-singular hyperbolic structure induced from M .

h.6. Dehn surges; on the figu_re eight knot.

m

For each value of w in the region R of the associated hyperbolic structure on eight knot, has

function d

Di2

manifold

, where K

is the figure

some Dehn surgery invariant d(w) = i(u(w) , A(w) ). The

is a continuous map from R

/ :1 of

element

S3 - K

I12

with vectors

v

to the one-point compactification

identified to -v . Every primitive

(p,q) of Z 9 Z which lies in the

(S3 - K)(P’q)

which

Actually, the map

d

image

d(R) describes a closed

possesses a hyperbolic structure.

/\L R

can be lifted to a map d

by using the fact that the sign of a

rotation of

(K3 -

z-ax

is)

I12 , is

well-defined.

(See §h.h. The extra information actually comes from the orientation of the z-axis determined by the direction in which the corners of tetrahedra wrap

around it) .

d

is defined by the equation

d (w)

(u , A) where

h. 18

uH(m) +

In order to compute the 2 and m

p.

h.ll.

two

for

image

(a rotation by + 2v)

d(R) , we need

first to express the generators

in terms of the previous generators

Referring to page

6, we

x and y on

see that a meridian which only intersects

2—cells can be constructed in a small neighborhood of K . The only

generator of iy

nl(P)

AH2(£)

wi(L(v))

(see p. h.ll) which intersects only two l-cells is

, so we may choose m = y . Here is a

figure eight knot can be arranged

cheap way to see what

1 is. The

(passing through the point at infinity)

so that it is invariant by the map

of

IR3 = S3 .

K)

v r—> -v

This map can be made an isometry of the complete hyperbolic structure

constructed for

53 - K

immediately from

Mostow's

. (This can be seen directly; it also follows theorem,

... ) .

an isometry of the Euclidean structure on

to

-1

. Hence,

a geodesic representing

This hyperbolic isometry induces

L(v) which takes m to m

and L

2 must be orthogonal to a geodesic

h.l9

representing m 'fi

, so

from the diagram on the bottom of p.

2 = +x + 2y is a longitude.

the curve

compute m

and

hill we deduce that

(Alternatively, it is not hard to

2 directly).

From p. h.l2 , we have

h.6.l

W(l

3(2.)

z2(l - z)2 d near the boundary of R

The behaviour of the map determine.

For

2

and

H(1)

is near the ray

w

instance, when

is near the ray

then

- z)

H(m)

is not hard to

Im(w) = O , Re(w) > 1 ,

Im(z) = 0 , Re(z) < 0 . The arguments of

H(m)

are easily computed by analytic continuation from the complete

3

case w = z = /:l (when the arguments are arg

Consequently,‘ (u , A)

H(m)

0

z

0 ) to be arg

is near the line

H(2)

: +2n

A = +1 . As w -» l we see from the

equation

z(l

- z)

w(l

- w)

=1

that

IZI2

-J

so then

1

(u , A) must approach the line u + kl = 0

IzI IwI2 -+ l

Then the map

lJ

IV]

d

.

, so (u , A) must approach

Similarly, as the line

u

w -+ +

- hi -

extends continuously to send the line segment 1 , fa

w

,

0 .

h.20

, +l ,

to the line segment

, +1 of the region R

There is an involution T

the solutions‘ z

and w

segment

so

u

follows that

d

to the line segment

When

IwI

we have also

is large and

,

u

- 2arg(w) . Thus - kl

=

0 , so

arg

d(ow) = ( u ,-A) . d

except when

In (W) 3_

H(m) =

(u , A)

w

or

if: 2

From. h.6.l

, -1 u/2 , then

arg w , arg

considering

IzI

is small

H(L) = 2n

IH(m)I

and

- harg w IH(£)I ,

= (h , 1)".

o of R which takes w to i—:_}

There is another involution

(and z to l_:—; ) .

,

0 < arg (w)
1

, and

in particular, the integer and half-integer (I) points along this ray, which

determine discrete groups.

The statement true fact that the six that at least

(since

ul(S3)

of the theorem is meant to suggest, but not imply, the exceptions do

223

have hyperbolic structures. Note

83 = (S3 - K)(l,0) does not admit a hyperbolic structure is finite.) We shall arrive at an understanding of the other

five exceptions by studying the way the hyperbolic structures are degenerating

as

(u , A)

h.8 Degeneration of

tends to the line segment

Merbolic

1 ,

,l

structures

h.8.l Definition A codimens ion—k folistion of an n—manifold M is a.

flLstructure, on

M

where

#7,

is the pseudogroup Of

local homeomorphisms

b.23

of JR

X

IIRk

which have the local form

¢(x

In other words,

.- 1')

(f(x

.y) .

g takes horizontal

z(v) )

(n—k)—planes to horizontal

(n-k)—planes. These horizontal planes piece together in M as

(n-k)-

(sub-manifolds), called the leaves of the foliation. M , like a book without its cover, is a disjoint union of its leaves.

For any pseudogroup

Nk

W

of local homeomorphisms of some k—manifold

, the notion of a codimension-k foliation can be refined:

h.8.2. Definit ion. An

J -structure for of

an-k >< Wk

Mn

, where

is a

is the pseudogroup of local homeomorphisms

which have the local form

¢(X with

J

W -foliation of a manifold M11

9

y)

(f(x

9

Y)

9

5(y) )

say If

then an

V’is the pseudo-group of local isometries of hyperbolic

71% -foliation shall, naturally, be

k—space,

called a codimension-k hyperbolic

foliation. A hyperbolic foliation determines a hyperbolic structure for each

k-manifold transverse When

w

w—simplex and the

to its leaves.

tends in the region RC!!! to apoint z-simplex are

both

IR

-

[0,1] , the

flattening out, and in the limit they are flat

1+. 2h-

If we regard these flat simplices as projections of non-degenerate simplexes A

and

B (with vertices deleted), this determines codimension

2 foliations on A and B , whose leaves simplexes:

are preimages

of points in the flat

u. 25 A and B glue together (in a unique way, given the combinatorial pattern) to yield a hyperbolic foliation on

S3 - K .

The reader should satisfy

himself that the glueing consistency conditions for the hyperbolic foliation near an edge result as the limiting case of the glueing conditions for the family of squashing hyperbolic

structures.

The notation of the developing map extends in astraightforward way

to the case of an

'%

-foliation on a manifold M, when

restrictions of a group J

’fib

is the set of

of real analytic diffeomorphisms of

N1‘ ;

it is

a map

D

:

Mn --->-Nk

Note that D is not a local homeomorphism, but rather a local projection map, or a submersion. The holonomy

H

1:1(M) ——>

J

is defined, as before, by the equation

D o T

H(c)

o D

Here is the generalization of proposition

3.6

to

$?-foliations.

For simplicity, assume that the foliation is differentiable:

h.8.1

are

Proms ition . if

J

E transitive

_

and if the isotropy subgroups ——_

compact , than the

developing map for any

‘gf-foliation

g;

J

x of a closed

1+.26 manifold M

is a fibration

D :

2522;; Choose

MP

-#

Nk

.

1k

a plane field

transverse to

32)

g2

greater than e we say that gl is greater then

; otherwise We

that this local ordering does not depend on our

l d(H(sl(x))

choice of y , we need to note that

U(gl , g2)

H(gg(x))

(in fact convex) set. This follows from

) < I } is a connected

= { x

,

the following lemma, the proof of which we defer.

14.9.l

~

:

51.82“”

f

=

d(glx , 32x)

' 1_S__8_._

convex function g H

One useful property of the ordering is that it is invariant under left

In other words g1

and right multiplication.

g3gl




G

—>

has

(id , id) since for any geG , (g , id) '—> id and

derivative zero at

(id , g)

[* , *] : G x G

1 . The tangent spaces of

tangent space to G

x G

at

G x id and id x G

span the

(id , id). Apply this to the group of isometries

ofHZ. From now on we choose a < I / 8 so that any two words of length four or less in which is the

GEGE

Ge

"smallest" element in ,a

7‘ B

id


id . In other words, if

, then a > B . This can be seen as follows. Take

x652

1.3

and look at its inverse image

and consider y

by the lifts of

1:) through y

and

and My) , where

a

a

under

GE

D . Choose

. We can

B (using the horizontal lifts of the geodesics

My) . Since this is a compact set there are

only a finite number of images of y there is one

which is

8

under

1rl

M contained in it.

Hence

8(y) whose IR co—ordinate is the closest to that of y itself.

8 is clearly our minimal element.

-

There are only finitely many translates of

y

in this region.

11-. 32

Now consider a > B > 1 , multiplication,

o‘lsa

e .




Then

a

>

and a >

> a

a‘lsa

> 1

1 then

>

s

invariance under left and right

By

o-lBa. so that >

1 .

>

1

oeo'l >

>

Suppose

“-1eas‘1 >

-1

s

.

1 so that 1 >

8.1 . Note that by multiplicative invariance, if 81 > g2 ggl = gilglggl > gllgaggl = gil . We have either 1 < 801—18—1u

aBa'IB-l or

So

>

a'lsa

Similarly if

then

:12

aEGE.

>

BcB-lo-l
PSL (2 ,

mfk

is a. polynomial map

(defined by

multiplication). Hence the dimension of the subvariety p = (l , ... , l)

is at least as great as the number of variables minus the number of defining equations .

I

We will later give an improved version of 5.2.2 whenever M has boundary components which are tori.

5.3.

In this section we will derive some information about the global

structure of the space of hyperbolic surface M

.

structures on a closed, oriented

This space is called the

W space

of M

and is

defined to be the set of hyperbolic structures on M where two are equivalent if there is an iSometry homotopic to the identity between them.

In order to understand hyperbolic structures on a surface we will cut the surface up into simple pieces, analyze

structures on these pieces,and

study

the ways they can be put together. Before doing this we need some information about closed geodesics in M

.

5.3.1 Pronosition: On

any closed hyperbolic

unique, closed geodesic

nsmanifold M

there is a

in any non-trivial

free

hmotogy

class. Proof: For any

ce_nlM

universal cover

En

consider the covering transformation

of M

. It

fixes some interior point of

En

is an isometry of

En

Tu

on the

. Therefore

it either

(elliptic), fixes a point at infinity

(parabolic) or acts as a translation on some unique geodesic (hyperbolic). That all isometries of

Proposition h.9.3

£235

He

are of one of these types was proved in

; the proof for

fin

is similar.

.A distinction is often made between

transformations in dimension 3

. In

"loxodromic" and "hyperbolic"

this usage a loxodromic transformation

means an isometry which is a pure translation along a geodesic followed by a non-trivial twist, while a hyperbolic transformation means a pure

translation. This is usually not a useful distinction from the point of view of geometry

and topology, so we will use the

Eggg_"buerbolic''

to

cover either case. Since

Ta

is a covering translation it can't have an interior fixed

point so it

can't be elliptic. For any parabolic transformation there are

points

moved

are

arbitrarily small distances.

This would imply that there

non-trivial simple closed curves of arbitrarily small length in M .

Since M

is closed this is impossible. Therefore

To

translates a

unique geodesic, which projects to a closed geodesic in M

. Two geodesics

5.8 corresponding to the translations geodesic in M

to the other.

To and To'

project to the same

if and only if there is a covering translation taking one

In other words

equivalently, c

(1'

=

g c

is free homotopic to

g_1

for some g e

1rlM ,

a .

5.3.2 Proposition : Two distinct geodesics EE universal cover M which are invariant by t at

or

covering translations have distinct

En 93 endpoints

a

Proof:

If two such geodesics had the same endpoint, they would be arbitrarily

close near the cammon endpoint . This would imply the distance between the

two closedgeodesics is uniformly i e. for all

5.3.3 W : I_n a Mex-belie homotogicalgl distinct represented by disjoint

M:

a , a

two manifold

M2

contradiction..

2 collection 2f_

d dis;oint non-trivial simple closed curves

E

, simole closed geodesics

Suppose the geodesics corresponding to two disjoint curves intersect.

Then there are lifts of the geodesics in the universal cover

32

which

intersect. Since the endpoints are distinct, the pairs of endpoints for

the two geodesics must link each other on the circle at infinity. Consider any homotopy of the closed geodesics in geodesics in

32 .

M2 .

It lifts to a homotopy of the

However, no homotopy of the geodesics moving points

only a finite hyperbolic distance can move their endpoints;

thus the images

of the geodesics under such a homotopy will still intersect,and this intersectic

projects to one on

M2

The proof that the closed geodesic corresponding to a simple closed

curve is simple is similar. The argument above is applied to two different lifts of the same geodesic.

Now we are in a position to describe the Teichmflller space for a closed surface. The coordinates given below are due to Nielsen and Fenchel. Pick

33

-3

disjoint, nonrparallel simple closed curves on

M2

(This is the maximum number of such curves on a surface of genus g .)

Take the corresponding into

of

2g

-2

geodesics

and cut along them . This divides

surfaces homeomorphic to

pants" from now on)

82 - three disks

with geodesic boundary.

M2

(called "pairs

is

--_‘\~

13/2.

23/3.

5.10 On each pair of pants P there is a unique arc connecting each pair of boundary components, perpendicular to both.

To see this, note that

there is a unique homotopy class for each connecting arc. Now double P along the boundary geodesics to form a surface of genus two. The union of the

two copies of the arcs connecting a pair of boundary components in P defines a simple closed curve in the closed surface. There is a unique geodesic in its free homotopy class and it is invariant under the reflection which interchanges the two copies of

P . Hence it must be perpendicular to the

geodesics which were in the boundary of

P .

This information leads to an easy computation of the Teichmflller

space of P .

5.3-h

:

f0”)

123 log 23 )

_i_s_ homeomogphi c

ED.

(log

21 ,

the

133 WW .

log

22 ,

with

where

co-ordinetes

ii = length 2:

The perpendicular arcs between boundary components divide P

into

two right angled hexagons. The hyperbolic structure of an all right hexagon is determined by the lengths of three alternating sides.

(See page

2.19)

The lengths of the connecting arcs therefore determine both hexagons so the two hexagons are isometric. Reflection in these arcs is an isometry of the hexagons and shows that the boundary curves are divided in half. lengths

1i / 2

The

determine the hexagons,; hence they also determine

P .

Any positive real values for the

ii

are possible so we are done.

I.

In order to determine the hyperbolic structure of the closed twomanifold from that of the pairs of pants, some measurment of the twist

with which the boundary geodesics are attached is necessary. Find 33

-3

more curves in the closed manifold which, together with the first set of

curves, divides the surface into hexagons.

In the pairs of pants the geodesics corresponding to these curves are arcs connecting the boundary components. However, they may wrap around the components.

In P it is possible to isotope these arcs to the perpendicula

connecting arcs discussed above.

moves

Let

2di

denote the total number of

o

degrees which this isotopy

boundary componat of p

.

the feet of arcs which lie on the

ith

5.12

Of course there is another copy of this curve in another pair of pants which has a twisting coefficient

are

gluedrtogether

di‘

. When the two copies of the geodesic

they cannot be twisted independently by an isotopy of the

Closed surface. Therefore

(di - di')

=

Ti

is an isotopy invariant.

If a hyperbolic surface is cut along a closed geodesic and glued

Remark

back together with a twist of

(n an integer) , then the

211' n degrees,

resulting surface is isometric to the original one.

However, the isometry

is not isotopic to the identity so the two surfaces represent distinct points in Teichmuller space.

Another way to say this is that they are

isometric as surfaces but not as markg surfaces. It follows that

1:i

is a well-defined real number, not just defined up to integral multiples

of

Zn .

5.3.5

W:

The TeichmflJler

space

J’(M)

genus g i_s homeomorphic t_o gg—ordinatas for

j(M)

£3 closed

m63-6

, (log

2.1

,

surface g

. There are M

11 ,

log

9.2 , 12

... , log

Ti

£38-3 , 138-3 )

where

ii

i§_the length and

Mara for a_system 22_

the twist

3g

-3

simple

closed geodesics

In order to see that it takes precisely 3g to cut a surface of genus g

Pi's

Therefore the number of has

3 curves, but

into pairs of pants

is equal to

- 2)

= 3g

Pi

-x(M8)

each curve appears in two

number of curves is 3/2 (2g

-3

Pi's

simple

closed curves

notice that

= 2g

x(Pi)

- 2 . Each

= -l .

Pi

. Therefore the

- 3 . We can rephrase Theorem

5.3

as

f(M) =

n‘3X(M)

It is in this form that the theorem extends to a surface with boundary.

cko)

The Fricke space

of a

surface M with boundary is defined to be the space of hyperbolic structures on M

such that the boundary curves are geodesics, moduloisometries isotopic

to the identity. A surface with boundary can also be cut into pairs of pants '

In this case the curves that were boundary curves

with geodesic boundary;

in M

have no twist parameter.

On the other hand these curves appear in

only one pair of pants. The following theorem is then immediate from the gluing procedures above.

5.3.6

:

3W)

_ii homeomoroh ic to

n'3X(M).

5.114 The definition of Teichmuller space can be extended

to

general

surfaces as the space of all metrics of constant curvature up to isotopy

and change of scale. In the case of the torus

set of all Euclidean structures zero) on

T2

with area one.

T2 ,

this space is the

(i.e. metrics with constant curvature

There is still a

closed

-

geodesic in each

free homotopy class although it is not unique. Take some simple, closed geodesic on

T2

and cut along it. The Euclidean structure on the resulting

annulus is completely determined by the length of its boundary geodesic. Again there is a real twist

glued to get

T2 .

parameter

that determines how the annulus is

Therefore there are two real parameters which

the flat structures on

T2 , the length

2. of a simple, closed geodesic

in a fixed free homotopy class and a twist parameter

5-3.T W :

1

along that geodesic

The ‘I‘eichmifller space if; the torus i_s homeomorphic _t_o_

Jam

withco

5-h

determine

Special algebraic properties

(log£,r),2,r

of a generating set

,5

g above

_or mus o_r" amiss

On large open subsets of PSL (2 , 0:)

132

e si-

, the space of representations

into PSL (2 , CI!) , certain relations imply other

relations. This fact was anticipated in the previous

section

from the

computation of the expected dimension of small deformations of hyperbolic

structures on closed three manifolds. surjective

(see

5.3.

The phenomenon that

dp

is not

) suggests that, to determine the structure of

5-15

n1M3 111M3

as a discrete subgroup of PSL (2 , II!) , not all the relations in

as an abstract group are needed. Below are some examples.

5.h.l

(Jorgensen) Let a , b ‘pgtwo isometries g

E gem fixed point g infinit . I_f w (,a , b)

(a'1 , b_l).=' 1.

such that w (a , b) = 1 than w and b

are

(i.e.

,g Trace

If a

(a) = rTrace (b)

PSL(2,m)) thenalso w(b,a) = 1.

i_n

are hyperbolic or elliptic, let 2 be the unique common

If a and b

Proof:

M

H3 with as. any word

perpendicular for the invariant geodesics

2a

1b

,

of

a

and b . (If

the geodesics intersect in a point x , 2. is taken to be the geodesic through

x perpendicular to the plane spanned by

of

a and b

2.5

and pass through

is parabolic,

b's

(say b is

fixed point at

2a

and

lb

) . If one

) 2. should be perpendicular to w

. If both are parabolic, 2.

should connect the two fixed points at infinity. In all cases rotation by l and b and b. , hence the first assertion. 180o in 2. takes a to

a’1

.If a and b with

invariant

are conjugate hyperbolic elements of PSL (2 , E)

geodesics

are perpendicular to midpoint between

Lb of

along

Eb

2. then m

6+1r/2 .

2.3

and

2b

, take the two lines m and n which

2. and to each other and which intersect and

2a .

Also, if

5b

2. at the

is at an angle of

should be at anangle of

6/2

and n

e

to

at anangle

516

Rotations of

versa.

Since

1800

a and b

through m

n

23

2b

to

and vice

are conjugate they act.the same with respect to

their respective fixed geodesics.

and

or n take

conjugate a _to b

It follows that the rotations about m

(and b to a) or a to

l

b-

(and b to

a—l) If one of a and b conjugate.

is parabolic then

they both are, since they are

In this case take m and n to be perpendicular to 2 and

to each other and to pass through the unique point X d (x , ex)

= d (x , bs) .

Again rotation by

180°

on in m

2 such that and n takes

atoboratob-l.. 1. This theorem fails when a and b fixed point

. For

example, consider

are allowed to have a common

5-17 where

A

*

...

(bk

I

Aer

. Then

A2

is chosen so that

1+

212

0

_

hke" bk

bk)2'

a.

is a root of a polynomial over Z , say

then a relation is obtained:

w(a ,b) =

-1)

l

However, w (a- , b

If

= I

in this caSe

(bah-l)2 =

(a)

.only if

f2

I .

is a root of the same polynomial.

This is not the case in the current example.

2. The geometric condition

that

and b

a

have a cannon fixed point at

infinity implies the algebraic condition that

solvable group.

5.h.2

a and b

generate a

(In fact, the commutator subgroup is abelian.)

mm W :

A_hy_ complete

huerbolic

manifold

fundamental 5393p i_s 51M 11pr elements involution

M3 m

a an_d b

admits an

s

v1 M2

i_s realized pygunigue

.

isomet

: Multiplication by an element in either fundamental group induces the identity map on the manifolds themselves so that

needs only to be

C)

defined up to composition with inner automorphisms to determine the isometry from

M1

to

M2 .

Since the universal cover of a hyperbolic manifold is

En

,

it is a

K (1: , l) . Two such manifolds are homotopy equivalent if and only if there is an isomorphism between their fundamental groups.

5.7.3

M1 and M2 are- hyperbolic manifolds which are complete with m volume, than they are homeomorphic i; d only

W

:

I;

_ii they are homotopy eguivalent . (The case gf_ dimension two _s well-known. )

For any manifold M , there is a homomorphism Diff M —* Out where

Out

(le)

automorphisms.

(le)

= Aut

Mostow's

/

Inn(1rlM)

the kernel

Diffo

is the group of outer

theorem implies this homomorphism splits, if M

is a hyperbolic manifold of dimension n

homorphism splits when

(nlM)

l 3 . It

is unknown whether the

M is a surface. When n = 2 [

(M) is contractible, provided

x(M) i 0 . If M is a

,

3-manifold which is not a Seifert fiber space, Hatcher has shown is contractible

5.7.h

:

;p_

J .

[

M11 pp Emerbolic

(nlM)

volume) and n 3_ 3 , then Out

pp_the gpoup

pg mum 23_

(complete , with finite total

Mn

.

finite gpoup, isomorphic

.

Proof : By Mostow's theorem any automorphism isometry of M

I”

Diffo M

that

|w

Haken

of

u M

induces a unique

1

Since any inner automorphism induces the identity on M ,

it follows that the group of isometries is isomorphic to

Out

(wlM)

Isom

Out

(le) .

That

is finite is immediate from the fact that the group of isometries,

(Mn) ,is finite. To see that

Isom

(Df3

is finite, choose a base point and frame at that

point and suppose first that M

is compact. Any isometry is completely

determined by the image of this frame

(essentially by

"analytic

continuation").

If there were an infinite sequence of isometries there would exist two image frames close to each other.

Since M

is compact, the isometries ,

corresponding to these frames would be close on all of

is homotopic to

¢2

outer automorphism If M

.

Since the isometry

on

“1M ,

p2-l

d’1

M . Therefore

the

, ¢ ,

induces the trivial

it is the identity; ie.,

is not compact, consider

¢l

submanifold

Mac

$2

=

o1 .

M which

consists of points which are contained in an embedded hyperbolic disk of radius

a . Since M has finite total volume,

Me

is compact. Moreover,

¢2

,

5:32 it is taken to itself under any isometry. The argument above applied to

Me

implies that the

group

of isometries of M

is finite even in the

nonscompact case. This result contrasts with the case n = 2 where

Out

(le2)

is

infinite and quite interesting.

The proof of Mostow's theorem in the case that

compact was completed by Prasad, and

5.7.2 (as well as

proved in Mostow,

J .

[

r

is not

Otherwise,

are

] . We shall discuss Mostow's

,

[

5.10 ,

5.7.1

giving details as far as they can be made

Later, we will give another proof due to Gromov, valid at

least for n =

5.8

/

generalizations to other homogeneous spaces)

proof of this theorem in geometric.

Hn

3.

Generalized Dehn surgery and hyperbolic structures

Let M be a non-compact, hyperbolic three—manifold, and suppose that

M has a finite number of ends

T2 x

El

, ... , Ek , each homeomorphic to

[0 , a) and isometric to the quotient space of the region in

(in the upper half-space model)

H3

above an interior Euclidean plane by a

group generated by two parabolic transformations which fix the point at infinity.

Topologically M

boundary is a union of

is the interior of a

T1 , ... , Tk

compact manifold

'M- whose

tori.

Recall the operation of generalized Dehn surgery on M

(section h.5);

5.33

(ai , bi)

it is parametrized by an ordered pair of real numbers

to

end which describe how

glue a solid torus to each boundary component.

If nothing is glued in, this is denoted by

82

be thought of as belonging to of

IR2

where

a

so that the parameters can

(ie., the one point

H1

(T2 , JR) ) . The resulting di = (a1 ,bi) or a. =

for each

compactification

space is denoted by

Mdl,..._,d.k

In this section we see that the new spaces often admit hyperbolic

d.i

= (ai , b.) structures. Since Mdl,...,dk is a closed man ifold when l 2 are primitive elements of (T , Z) , this produces mauy closed .

_

H1

First it is necessary to see that small deformations

hyperbolic manifolds.

of the complete structure on’ M

space M

5.8.1

d1

’ a]:

,

m

U

: Apy small defamation

T2X[0,l]

extends

determined pp 32 sign operators

M:

induce a hyperbolic structure on some

o,B

9;g"standard'' Mubolic structure 2

ppsome(D2xsl)d. py

of

the traces

an2 .

A "standard" structure on

T2 x

g the

d=(a,h)_i_s_

matrices representing

[O , 1] means a structure as

described on an end of M truncated by a Euclidean plane. The universal

cover of

T2 x

[0 , l] is the region between two horizontal Euclidean planes

(or horospheres), modulo a. group of translations. If the structure is

deformed

slightly the holonomy determines the new structure and the images

5.3h of

a

and

8 under the

If H(o)

is still parabolic then so is H(B)

equivalent to the standard one.

axis

2 in

H3

H are slightly perturbed.

holonomy map

OtherwiSe

and the structure is

H(c) and H(B) have a common

. Moreover since H(c) and H(B) are close to the original

parabolic elements, the endpoints of

2 are near the common fixed point

T2 x

[0 , l] is thought to be embedded

If

of the parabolic elements.

in the end,

towards image of

a

T2 x

.)

[o ,

, this means that the line lies far out

and does not intersect

T2 x

[0 , l] in

H3

T2

[0 , l] . Thus the developing

X

for the new structure misses

be lifted to the universal cover

H3 -

N2

of

H3

2 and can

-2

This is the geometric situation necessary for generalized

surgery. The e

{fi(a)

.3(3)}

ion to ,

whet e

3

(D2 x 81)d

is just the completion of R3

is the lift of

H to the cover r\/

that the completion depends only on the behavior of H(u)

2 . In particular, the pair

if

H(

a

d = (a , b)

H(m) +

N

53—17. r\/

and H(B)

- 2.

/

Recall ‘

along

) denotes the complex number determined by

(translation distance along 2 , rotation about

Dehn surgery coefficients

Dehn

b

2) , then the

are determined by the formula:

H(B) =

r2ri

The translation distance and amount of rotation of an isometry along its fixed line is determined by the trace

of its matrix in PSL (2 , m) .

This is easy to see since trace is a conjugacy invariant and the fact is

5-35

clearly true for a diagonal matrix.

In particular the complex number correspondi

to the holonomy of a matrix acting on its trace.

I

H3

is log A

where

The main restflt concerning deformations of M

5.8.2W : E

M =

,. admits

idk )e U

£351; (dj-S

hnerbolic

M0

manifold

A-1

is

is

a pyperbolic

structure then

d1""’dk

M

S

structure.

Consider the compact submanifold

Proof:

end.

I.-

A +_

hborhood ugh,“ . ,w)i_nSZXS2X.-..x82

thereisg nei such that

M. ’

.

Mo: M

gotten by truncating each

has boundary a union of k tori

and is homeomorphic to the

M

By theorem

such that M

= interior

M

.

5.6

,

MD

has a k

complex parameter family of non-trivial deformations, one for each torus.

From the lemma above, each small defamation gives a hyperbolic structure on some

Mdl ,

neighborhood of

, ‘11; . It remains to show that the (an ,

di

vary over a

, an) .

Consider the function

Tr: Def (M) —>

(Tr ( H(aln

.

.Tr (H(akn)

which sends a point in the deformation space to the k-tuple

of the holonomy of

cl , a2

fundamental group of the

iEll

,

, ak ,where

ai

,

Bi

of traces

generate the

torus. Tr is a holomorphic

(in fact,

lenWI q

5.36 algebraic) function on the algebraic variety Def LM) . Tr

Note that

(:2 , ... , 12) for some fixed choice of signs. if and only if if the

iEE-

H(ci)

di

surgery coefficient

... , k

equals

Ma

the hyperbolic structure on for i = l ,

3

H(ai)

in terms of

and

H(Bi)

,

Man

Ho (ulT) ,

:2

Tr-1

w

, :2)

(i2 ,

it follows from

(:2 , ... , i2)

iEQ-

torus depend on the

H(Bi)

it is necessary to estimate

in order to see how the surgery coefficients vary.

Restrict attention to one torus T image of

Therefore di =

is unique.

is an open neighborhood of

of both H(ci)

(H(ai) =

Tr of a small open neighborhood

Since the surgery coefficients of the

trace

Tr

. By Mostow's Theorem

Since dim (Def (M) ) 3_ k

that the image under

, ... , a

a

)=

.

is parabolic if and only

only in the original case and

consists of exactly one point.

of M

H(ai)

is parabolic and

(M°° , ... , a

and conJugate the original developing

, ... , a so that the parabolic fixed point of the holonomy,

is the point at infinity.

to put the holonomy matrices of the the following form:

no_(o)

=

Note that since

generators

[1 1] 0

Hobs)

By further conjugation it is possible

l

. 110(8)

a

no (8)

,

B

«IT

of

in

[l c] 0

1

act on the horospheres 3‘00“ " as

a two dimensional lattice of Euclidean translations, c and l are linearly

5.37

Roux)

IR . Since

independent over

,

30(8)

[1]

is near

Since 3(a)

0

Ho(c)

|: :|

L]

1

2

l

_

1

a

A

2

—

31c) =

between

c

1) I(A1“ - 15

’

Since

u

) respectively.

1 -1

e2

Therefore

Similarly,

2 Z

1

a

However

A

_AE-l *

(11

l

=

l

a

Let the

and

,

0

H(m)

as an

S

(A , A ‘1) and

3(a) and H(B) be

eigenvalues of

[1

, say

0

H(m) , H(B) will have

eigenvector, the perturbed holonomy matrices

common eigenvectors near

[J]

have

log. A

3(a) and 318)

v and H(B)

l

For

A

A

A'1 —l

u

= log 11

-n

, u

near

1

c

-

The surgery coefficients

(a. , b)

-

this is the desired relationship

are determined by the formula

5.38 a

Eh.)

+ b

313)

:2

'n'

i

From the above estimates this implies that (a

+

log A

b c )

(Note that the choice of sign corresponds to a choice of A

Since 1

c are linearly independent over

and

vary over an open neighborhood of of

l . Since Tr (H(m)) =

(up to sign) in the

image of

°° as

A + A _1

A

IR , the values of

.) (a. , b)

varies over a neighborhood

varies over a neighborhood of

mk Mdl’”"dk

Tr : Def (M)

that the surgery coefficeints for the

A4L

or

2

, we have shown

—>

possessing

hyperbolicrstructures vary over an Open neighborhood of

a

in each component.

The complement of the Borromean rings has a complete hyperbolic

structure. However, if the trivial surgery with coefficients performed on one component, the others are unlinked.

M 0 , we will study the decomposition M .

M(O,5]

consists of those points in M

5.5

non-trivial closed loop of length

, and

through which there is a

M[e

9

a) consists of those

points through which every non-trivial loop has length

In order to understand

the geometry of

universal covpr

M

En

and any x e

En

of

T which move x a distance i_s , and let

=

Hn

let

= M( O,e] U M [5,“),

M(0

2.5 .

,3] , we pass to the

. For

any discrete group

Fe(x)

be the subgroup generated by all elements

subgroup consisting of elements

whose derivative

P of isometries of

ré(x) 0

2f_ isometries g; H21

P

PE(x)

and for evepy

has pp_ abelian subgroup 93_ finite index.

This proposition is much more general than stated; if

is replaced by

"abelian"

"nilpotent" , it applies in general to discrete groups of

isometries of Riemannian manifolds with bounded curvature. The proof of the general statement is essentially the same.

In any Lie group G , since the commutator map G x G -» G has derivative 0 at

4*

[ ,

J

(l , l) , it follows that the size

].1

Il1J.

i !'1

5.52 of the commutator of two small elements is bounded above by some constant times the product of their sizes. Hence, if

r;

is any discrete subgroup

G generated by small elements, it follows immediately that the lower

of

1'"a

central series

3 [1" , I"_]D[I"

s

e

s

, II”a , P'JJ a

,

there is a lower bound to the size of elements of

F;

is nilpotent.

(by

idea is

first

Fe(x)

to find an

P;

is actually abelian.

has an abelian subgroup of finite index, the

I; (x)

s such that l

1 then choose a many times smaller than £1 , of

r€(x)

will lie in

) . In other words,

considering, for instance, the geometric classification

of isometries) that this implies

To guarantee that

P;

G is the group of isometries of hyperbolic space,

When

it is not hard to see

is finite (since

P;

1

SO

is always

abelian, and

the product

of generators

(x) . Here is a precise recipe:

Let N be large enough that any collection of elements of 0(n) with

a_N contains at least one

cardinality

El

more than

Choose of

H11

at x

61 /

6

/ 3

32 §_

5

1 / 3 so that for any pair of isometries

which translate a point x

of

pair separated by a distance not

¢l o ¢2

a

distance

5.32 ,

¢l

and

¢2

the derivative

(parallel translated back to x) is estimated within

by the product of the derivatives at x

of

pi

and

¢2

(parallel

translated back to x) .

Now let translating

a

x

=

52

2N

a distance

so that a product of 2N isometries, each

5.5 ,

translates

x

a distance

5.32 .

LEt

5-53 g1 ,

, gk be the set of elements ofI‘ which move x a distance

is ; y

e

I'e(x)

gil

=

bewritten

< N

-

in the generators

gig,

...-

(GM-1 ' (a

e‘ B) =

rél (x) . By I'él(x)

group

rél (x) ,

so

l

(x) , where

, gk . In fact, if

:N

e'

(c , s' , B#l)where

' 8.,

61/ 3

a' is within

B-l e' B

is in

induction, the claim

of

I‘E': (x) ;

1:

gi's

in the

'

réiX) n re (x)

, it can

B

has length: N

l . It follows that

hence

verified.

with finite index.

Y

the coset

I‘;

(x) =

abelia:

Thus, the

has finite index in the group generated by

I‘e(x)

and

I

n = 3 , the only possibilities for discrete abelian

When

Examples

gl ,

is any word. of length

e'

y= o-

and the derivative of

(as)

I"';

. Consider the cosets y

; the claim is that they are all represented by y's which are

words of length y

I‘€(x)

they generate

Z (acting

groups are Z (acting hyperbolically or parabolically) , Z x

parabolical-ly, conjugate to a group of Euclidean translations of the upper half-space model),

rotations of

Z

x

some axis) ,

Zn

(acting as a group of translations and

222

and

X

Z2 (acting by 180°

rotations

about three orthogonal axes). The last example of course cannot occur as

I‘; (x)

.

Similarly, when

T;(x)

occur as

is small compared to

i,

Z x

Zn

cannot

.

.

Any

a

discrete

group

n-l

I' of isometries of Euclidean space E

as a group of isometries of

Hn

, via the upper half-space model.

acts

5.51;

For any x

5.10.1

Thus, that

sufficiently high

r

contains as a special case one of the Bieberbach

contains an abelian subgroup of finite index.

Fél(x)

Fe(x) n

is parabolic,

this, note that if of y

(in the upper half space model) ,

could lie in

Fe(x)

Pél(x)

Pe(x)

Pe(x)

theorems,

Conversely, when

must be a Bieberbach group. To see

contained any hyperbolic element 7 , no power

, a contradiction. Hence,

re(x)

must

consist of parabolic and elliptic elements with a common fixed point

at

a

at p

=r

p

, so it acts as a group of isometries on any horosphere centered

.

P;

If re(x)r) (x) is not parabolic, it'must act as a group of l . translations and rotations of some axis a . Since it is discrete, it ,

contains

Z with finite index provided

follows that

re(x)

I‘e(x)

is infinite).

It easily

is either the product of some finite subgroup F

of

5.55 0(n

- l)

(acting as rotations about a) with Z , or it is the semi-

direct product of such an F with the infinite dihedral group,

Z/2 * z /2

The infinite dihedral

H3

group acting on

Foranyset

SCHn,let Br(S)

5.10.2 W . 3

hyperbolic

{xéHn|d(x,S) 1r}.

_

There jig a > 0 such that for 2y

- manifold

——

—

M , each copponent of M

(0.5]

(1)5 horoball modulo mgzzez,g_ (2)

Br(g)

modulo

Z, where

complete

oriented

——-— is either

g i_sg geodesic

The degenerate case r = 0 may occur . Proof.

Suppose x

eM(03 s] .

Let x e

to x . There is some covering translation y

i e . If

y

is hyperbolic, let

translations along

H3

be any point which proJects

which moves

at

a distance

a be its axis. A11 rotations around

a ,

a , and uniform contractions of hyperbolic space along

.

5.56 orthogonals to a commute with

Br(a)

,

a

Hence

. It follows that

, where r = d (a , x) , since Y moves any

a distance

at

y

_ X .

Gromov's norm on the real singular homology (really it is only a pseudo-norm) is obtained from this norm on cycles by passing to homology: is any homology class, then the norm of

the norms of cycles representing

6.1.2

01

if

= inf

{”le

,

I

z is a singular cycle representing

a}

It is immediate that

and for AeIR,

If

f : X —> Y

Hk(X; IR)

a is defined to be the infimum of

First definition.

”a”

a e

”OMB“ 5 ”all + ”Bil “Doll s D»! ”a”

is any continuous map, it is also immediate that

6.2

“mall

6.1.2

In fact, for any cycle

2

“all

_ M2

I! [Ml] H 2 Ides fl H [M21H

6.1.3.

invariant of M

,

.

What is not immediate from the definition is the existence of any nontrivial examples where

H

The n-sphere

n

Example.

[M] H

21

7! O

.

admits maps

where

2:.L

same

uziu

is

=

(l/i)

{zi}

1/1,

0'1

n

S:1 —> Sn

of degree 2 (and

[Sn] H = O . More explicitl , one representing the fundamental class of S:L ,

higher). As a consequence of 6.1.2, may picture a sequence

f :

0'].-

and

H

wraps a l-simplex i times around

H

[so] H

=2 .

Consider now the case of a complete hyperbolic manifold

0'vo’ ...,vk

vo,...,vk

k : A

—>I-II1

.

[shit = 0.

As a trivial example,

k + 1 points

Sl

in

,

Mn = fin .

determine .

M11

. Any

a straight k-simplex

whose mage 15 the convex hull‘I of

vo,..

There are various ways to define canonical parametrizations for

,vk

0'vo,...,vk

6.3

here is an explicit one.

vo,... ,vk

In this model,

a .

affine simplex

2'.

ti vi

IRn+l J

Consider the quadratic form model for

so they determine an

[In barycentric coordinates, on

. The central projection from .

-_ xl2 +

0'v0’...,vk

in

‘2 _ -1 xn - xn+l —

+ 2

HI1

in

M = fin

, since

Ak

.

gives

(to,... ,tk)

straight

'7':

back to one sheet of the . . a parametrized straight

H1’1

tural with respect to isometries of

—> M

1

can be lifted to a singular simplex

is simply connected.

straight simplex with the same vertices as

projection of

or

O of

fly.

Q

Any singular simplex

7r,

(§2.5).

. This parametrization is natural with respect to affine maps of

hyperb0101d simplex

an+l ,

become points in

Hn

7r

Let straight

and let straight

(7:) (1')

be the

be the

back to M . Since the straightening operation is natural,

(1') does not depend on the lift T .

Straight extends linearly to

a chain map

C*(M) —> C*(M) ,

straight :

chain homotopic to the identity.

(The chain homotopy is constructed from a

canonical homotopy of each simplex

for

any chain

c

,

”straight (c)”

'r

to straight

S ”c”

.

(1) . ) It is clear that

Hence,

in the computation of the

norm of a homology class in M , it suffices to consider only straight simplices-

6.1.h. 'Prcnosition. There is a finite suprexmm volume

Proof.

of a

vk to the k-dimensional

straight k- simplex in hyperbolic space

Hn

provided

k

#

It suffices to consider ideal simplices with all vertices on

since any finite simplex fits inside one of these.

l.

S°° ,

For k = 2 , there is only

one ideal simplex up to isometry. We have seen that 2 copies of the ideal triangle fit inside a compact surface

which equals

(§3.9). Thus

it has finite volume,

n by the Gauss-Bonnet theorem. When k = 3 , there is an

efficient fonnula for

the

computation of the volume of an ideal 3-simplex; see

Milnor's discussion of volumes,

6.7

The volume of such simplices attains

its unique maximum at the regular ideal simplex, which has all angles equal

I

6.1.5

3.1hl5926...

n

. Thus we have the values

4

600

4

to

1.0lh9hl6...

It is conjectured that in general,

vk

=

fl

is the volume of the regular ideal

k-simplex; if so, Milnor has computations for more values, and a good asymptotic

formula as k -—>

w

can be obtained for

. In lieu of a proof of this conjecture, an upper bound

vk

from the inductive estimate

6.1.6

vk-l

To prove this, consider any ideal k-simplex one of its vertices is the point at U

m

G

in

HR .

Arrange

6

so that

in the upper half-space model, so that

looks like a triangular chimney lying above a k-l face

do

of

U

.

6.5

de de=

kthe Euclidean

(—)k de.

Let

be

Let

volume element, so hyperbolic volume is

denote the projection of

1

k

h(x) denote the Euclidean volume of

v(0') =

0' is

height of

f foot-k

Euclidean k-l volume element for

(k-l) v M2

a

, we

obtain

6.2.

i_s any map between c_105ed oriented hyper-

bolic n-manifolds, then

v(Ml) 2 Ides fl v(Ma) Gromov's theorem can be generalized to any (G,X)-manifold, where G acts transitively on

X with compact isotropy groups.

To do this, choose an invariant Riemannian metric for X and normalize Haar measure on G as before. The smearing operation works equally well, so that one has a chain map

smearM

:

:kOC) —>

k(M)

-

In fact, if N is a second (G,X)-manifold, one has a chain map

smearmM : :kU‘T) —> 5km)

:

defined first on singular simplices in N via a lift to X linearly to all of

smearN,M(z)

(IAN) .

represents

If z is any cycle representing

(v(N)/v(M) )[M] . This

!| [N] v N

Interchanging M

and N

6.2.2 Theorem. For

any

, and

then extended

[N] , then

gives the inequality

II VI II [M] H W)—

, we

obtain the reverse inequality, so we have proved

pair

(G,X) , where G acts trans itively on X

compact fl—lisotropy groups and for

any invariant volume form

with

% X , there E a

6.10

constant C

such that every closed

H

(G,X)-manifold M satisfies

0riented

[M]

I

= C V(M) ,

I

(where v(M) is the volume of M) .

This line may be pursued still further. In a hyperbolic manifold a smeared k-cycle is homologically trivial except in dimension k = 0 or k = n

(G,X)-manifolds when G does not

but this is not generally true for other

act transitively on the frame bundle of‘ X . The invariant cohomolog

HEM)

is defined to be the cohomology of the cochain complex of differential forms

on

X. invariant by G . If 0: is any invariant cohomology class for X

it defines a cohomology class

C(M

on any (G,X)-manifold M . Let

,

PD(7)

denote the Poincare dual of a cohomology class 7

6.2.3

Theorem. There is a. norm

(G,X)-manifold M

oriented

H II

H:(X)

such that for any closed

a

ll MGM) I PI‘oof.

o_n

= v(M)

”all

It is an exercise to show that the map

smearM,M : H*(M) —> H*(M) is a retraction of the homology of M

in M

of

Rig-(X) .

to the

Poincare’ dual

of the image

The rest of the proof is another exercise.

I

In these variations, 6.2.2 and 6.2.3, on Gromov's theorem, there does not seem to be any general relation between the proportionality constants and

,

6.11

the maximal volume of simplices. However, the inequality

6.1.7

readily

X possesses an invariant Riemannian metric of

generalizes to any case when

non-positive curvature .

6.3

Gromov's proof o_f Mostow' s theorm . Gromov gave a very quick proof of Mostow's theorem for hyperbolic

3-manifolds, based on

6.2. The

proof would work for hyperbolic n-manifolds if

it were known that the regular ideal n-simplex Were the unique simplex of

maximal volume.

6.3.1.

The proof goes as follows:

I_f M1 and

manifolds , then

M2

g homotog equivalent , closed, oriented

O . There are

oriented ideal

0' such

spanned by the image of its neighborhoods of the vertices

in the disk such that for any simplex 0" with vertices in these

neighborhoods,

v(straight(FlU') 5 v3 - 8/2 .

Then for every finite simplex

6.12

0'0

very near to

0'

isometric copies

0'

Such a simplex

0';

,

this means that a definite Haar measure of the

0'0

of

near

0"

can be found with volume arbitrarily near

then the "total volume" of the cycle

exceeds the total

v(straight(flo'c'>)) < v3 - 8/2.

have

volume

of

z=

35-

straight(f*z) ,

smear

v3

(0'; - 66-)

contradicting

. But

strictly

6.3.1.

To complete the proof of Mostow's theorem in dimension 3, consider any ideal regular simplex

0'

together with all images of

repeated reflections in the 'faces of images of

0' is

vertices of

F1

0'

a dense subset of

, it

0'

s: -

7T1 M1

Once

Fl

is known on three of the

is determined on this dense set of points by

to the action of

from

. The set of vertices of all these

s:

must be a fractional linear transformation of

action of

0' coming

‘rrl M2

,

6.3.2,

so

conjugating the

'

. This completes Gromov's proof

of Mostow's theorem.

In this proof, the fact that

that v(Ml) = v(ME)

is a homotopy equivalence was used

and (b) that

?1

extends to a map of

With more effort, the proof can be made to work with only assumption

6.1+ Theorem. (Strict version

E degree 74 0

Gromov's inequality

f :

(a)

.

:

Ml —> M2

b_e

between closed oriented hyperbolic 3—manifolds such that

6.2.1 E equality, i.e.,

v(Ml) = [deg fl v(Mz) Then f i_s homot Ohio t_oanfipwhich

.

Ea local 1somet

.

if: [deg fl

I

any map

of Gromov's theorem) . Let

s:

P

to show (a)

fl

f is a homotog equivalence and otherwise it is homotopic to a covering rug.

6.13

Pr_oof_. The first step in the proof is to show that a lift universal. covering spaces extends to thesis of

6.1-I- has to

SE -

F

of

f to the

Since the information in the hypo-

do with volume, not topology, we will know at first only

that this extension is a measurable map of

Si .

Then, the proof in

6.3 will be

adapted to the current situation.

The proof works most smoothly if we have good information about the asymptotic behaviour of volumes of simplices.

in

H3

a. regular simplex

all of whose edge lengths are E .

6.1+.l. The volume

9:

0'E

differs from the maximal volume

which decreases exponentially with

Proof o_f 6.1L.l.

:0 6 1-13

O'E

xo

by drawing the four rays from a point

0’“

these rays, a distance D

C be the distance from

dv(UE)

‘

a face of

centered at

x0,

from

1:0

x0

centered at a point through the vertices of an

. The simplex whose vertices are on

is isometric to

0E

for some E . Let

to any face of this simplex. The derivative

is less than the area of 0'E

v3 L a quantity

E .

Construct copies of simplices

ideal regular simplex

x0

O'E be

Let

5 GE

times the maximal normal velocity of

. If on is the angle between such a face and the ray through

, we have

(NOTE) 'dD

From the hyperbolic law of

dv(oE) dD

.

Sines

(2.6.16)

decreases exponentially with

corresponding statement for E

< 21r

5111

sin

a

a sinh C = sinh D

.

showing that

D (since sinh C is bounded) . The

follows since asymptotically, E ~2D + constant.

6.1M

Clara

6.h.2.

E simplex with

volume close

t_o

v3

has all

dihedral angles close t_o

60° . Proof o_f

6.11-.2. Such

a simplex is properly contained in an ideal simplex

with any two face planes

follows from

6.1+.3. near

V

the same, so with one common dihedral angle.

7.

There is some constant C

3

6.h-.2

andforany angle

such that for every

simplex

0' with volume

Bonafaceflc,

v3-v(0') 3 c 62 Proof of

6.11-.3.

If the vertex v has a face angle of

that the other three vertices are at

on

The new spike added to

0'

enlarge

0

so

, without changing a neighborhood of 2

v . Now prolong one of the edges through v to this edge.

B , first

beyond

.

Sea , and push

v

out along

v has thickness at v estimated

6.15

by a linear function of

B

quadratic function of

B (from 2.6.12),

so its volume is estimated by a

. (This uses the fact that a

cross-section

of the spike

is approximately an equilateral triangle).

6.1+.3 6.1+.l+. For

fl(r)

every point

converges t_o _a point

6.1+.l-L. Let

Proof o_f

let the simplex

at

x0

x0

O'i

x0

e

in

on

s:

H3 , and let

r throgh x

r be some ray emanating from

x0

(with all edges having length i ) be placed with a vertex

and with one edge on r

in a neighborhood of

M1 ,_aii almost every ray

x0

,

Ti

and let

6i

be a simplex agreeing with

r lengthened, to have length

but with the edge on

i+l.

0’;

61

1"“

\\

x

an amount

Si

0'1

and

3.3..

Ti D O'i

decreasing exponentially with

i

, so

..

r

and

straight

f*

[Ml] .

smeaerUi

and

smearM

U

l

Since straight

f*

must also be very efficient. In other words, for all but a set of

5 v(Ml) 8i

of simplices

v

straight

smearM 1i l

v(Ml) = ldeg fl v(Ma) , the cycles 1 measure

“......“ "llla

deviate frOm the supremal volume by

are very efficient cycles representing a multiple of

smearM Ti

[1

...-

AD The volume of

'5

l

3

fO' must have volume

0'

2 v3

in smear

- Si

0'i

(or smear

ti)

, the

simplex

6.16 Let 'B be a ball around

x0

M1 . smearM 0"i

which embeds in

smearB 1i correspond to the measure '

for

The chains and

to those singular simplices with the first vertex in

which take

xo

to B , all simplices

mapped to simplices straight

v

I(0'i)

smearBO'

f

smearM Ti

thoeo image

Z

2v (Ml)

Thus for all but a set of measure at most

3

smearB (Ti

Si

. restricted

of B

M1

in

of isometries I

i=i

I(Ti) for all volume 2 v3 - 8i

and

with

and

> io

i

are

. By 6.1+.3,

the sum of all face angles of the image simplices is a geometrically convergent series.

It follows that for all but a set of small measure of rays r

from points in B

io —> on , f(r)

, f(r) converges to a

s:

point on

it follows that for almost every ray

r

emanating

; in fact, by letting

emanating from points in

B ,

x'

Then there must be a point in B such that for almost ' every ray r emanating from x , f(r) converges. Since each ray emanating from converges.

a point

in

H3

~

I

through all points in

H3

6.1m.

.

-

Remark This measurable extension of f, to

5:

actually exists under very general

circumstances, with no assumption on the volume of

that if g

, this holds for rays

is asymptotic to some ray emanating from x

is a geodesic in

M1 , RS)

M1

and

M2 .

The idea is

~

M2

behaves like a random walk on

Almost every random walk in hyperbolic space converges to a point on

82-1

. .

(Moral: always carry a map when you are in hyperbolic space!)

6.’+.5

The measurable extension o_f_

every positively oriented ideal

positively oriented

Proofo_f6.l+.'-". I

ideal re

Let

t_o

regular

Si

simplex

_

carries the vertices

93 almost

to the vertices g another

ar simplex.

Consider a pOint

embeds in M , as before.

F

0'1

1:0

in

H3

and aball B about

be centered at

xO

xO

which

As before, for almost

6.17

all isometries I which take has volume converging to

xo

{straight f° I ° 0'1}

to B , the sequence

v3 , and all four vertices converging to

SE

.

If for almost all I these four vertices converge to distinct points, we are done. Otherwise, there is a set of positive measure of ideal regular simplices such that the image of the vertex set of

0' is degenerate: either all four

vertices are mapped to the same point, or three are mapped to one point and the

Q

third to an- arbitrary point.

Q

We will show this is absurd.

If the degenerate

v0 and vl

cases occur with positive measure, there is some pair of points with

f(vo) = f(vl)

vo, v1, v2, v3 set A

,

such that for almost all regular ideal simplices spanned by

either

f(va) = f(vo) f(A

of positive measure with

ideal simplex with two vertices in A

or

f(v3) = f(vo)

a single point.

under

'ITl M2 _Pr_o'c£

6.l-I- which

5E 6.1+.

(One method is to use ergodicity

will follow) . The image point

covering transformations of

has a fixed point on

Almost every regular

has one other vertex in A . It is easy

to conclude that A must be the entire sphere. as in the proof of

. Thus, there is a

S“ ,

It follows from

M1 .

f(A)

This implies the image of

which is absurd.

6.’+.5 that

v0, v1, v2,

Trl M1

6.h.5

there is a vertex

almost all regular ideal simplices spanned by

is invariant

vo

v3 ,

in

l

such that for the image vertices

6.18

span a regular ideal simplex. Arrange

v0

vo

the plane,

E2 .

f(vo)

to be the point at

Three other points

infinity in the upper half-space model.

an ideal regular simplex with

and

v3

v1, v2,

span

iff they span an equilateral triangle in

By changing coordinates, we may assume that

f maps vertices

of almost all equilateral traingles parallel to the x-axis to‘ the vertices

In

of an equilateral triangle in the plane.

complex notation, let

to

=

span and equilateral triangle. For almost all z e

so that O, l, w

entire countable set of triangles of the form spanned by vertices

-k

-k

z + 2 (n+1) , z + 2 (n+w)

[k,n,m e Z]

z+

3a

(1'.

, the

2-kn ,

are mapped to equilateral triangles

-11/

Then the map

? must

take the form

~f(z + 2'k(n+mw))= g(z) + h(z) for almost all z

. The function h is invariant a.e. by the dense group T

of translations of the form

h

is

2'k(n+mw)

2

constant a.e. Similar

—> z +

2-k(n+mw)

. This group _is ergodic, so

reasoning now shows that

g

is

constant a.e.,

so that f is essentially a fractional linear transformation on the sphere ‘

3:

Since

Isom(I-13)

'1‘".Ta

, to a

Tf*oz ° '5,

subgroup of

11'1M2

this shows that

TrlMl

is conjugate, in

6.1+

I

6.19

6.5

Manifolds with haunting.

There is an obvious way to extend Gromov's invariant to manifolds with boundary, as follows.

the relative chain group

The norm on element of

:k(M) tk(M,A)

If M

fk(M,A)

‘

'y e

Hk(M,A)

norm on ka(M,A)

goes over to a

is the total variation of

u

: the norm

Z:(M)/ Z:(A) . ll ull

of a

restricted to the set of

ll 7”

of a

homology

is defined, as before, to be the infimal norm of relative

7

cycles representing

with boundary

submanifold,

is defined to be t e quotient

singular simplices which do not lie in A . The norm

class

CM a

is a manifold and A

.‘

Gromov's invariant of a compact, oriented manifold

(M,8 M) is

ll

ll

[M,5 M]

, where

[M,5 M] denotes the relative

fundmental cycle.

There is a second interesting definition which makes sense in an important special

case. For concreteness,

we shall deal only with the case of 3-manifold

whose boundary consists of tori. For such a manifold M

ll

[M,B M]

Observe that

”o = a—>O lim inf {llzll

a2

l

z represent, [M,5M] and

represents the fundamental cycle of

condition for this definition to make sense is that

BM

in the present situation that self-maps of degree

> 1 . Then

, define

B M , so

ll [8 M] H

”6le S a} that a necessary

= O . This is true

consists of tori, since the torus admits

ll (M,8M) ”o

is the limit of a

non-decreasing

sequence, so to insure the existence of the limit we need only find an upper bound.

This involves a special property of the torus.

6.5.1.

Proposition.

trivial cycle

E

There _i_s_

{2(T2) ,

a constant

then

2

K such that _if z _i_s afl homologioag

bounds _a_chain

c with

”CH SKllzll

.

6.20

Proof.

Triangulate

Partition

T2

T2 (say, with two "triangles" and a single vertex) .

into disjoint contractible neighborhoods of the vertices.

Consider first the case that no simplices in the support of

2

have large

diameter. Then there is a chain homotopy of z to its simplicial approximation

a(z)

The chain homotopyhas a norm which is a bounded multiple of the norm of z . Since simplicial singular chains form a finite dimensional vector space, a(z) is homologous to zero by a homology whose norm is a bounded multiple of the norm

of

a(z) . This gives the desired result when the simplices of z are not

large.

finite

In the general case, pass to a very large cover

sheeted covering space

transfer: {*(M) —> KAI?) average of its lifts to simplices.

Clearly

is any cycle on

the transfer of

transfer

2

p

p :

M¥> M

T2

of

T2 .

For

any

there is a canonical chain map,

. The transfer of a singular simplex is simply the

if; this extends

otransfer = id

in an obvious way to measures on singular

, and

lltransfer

c

ll

2

llcll

. If 2

T2 , then for a. sufficiently large finite cover 352 of T2 , z to T2 = T2 has no large 2-simplices in its support. Then

is the boundary of a chain

c with

He”

__

C

-

,

V

5.11.2

k . This follows from the analysis in

components of

M[e

a)

k

such that every manifold with

cusps has volume

: the number of boundary

is bounded by the number of disjoint

e/2 balls which

I

can fit in M . It would be interesting to calculate or estimate the best constant C .

6.6.3..W . class

Q connected

The set g values

g Gromov's

invariant

“[

]

“O

o_n the

manifolds obtained from Seifert fiber spaces and complete

Marbolic manifolds g finite volume

Ea closed well-ordered We shall see later

subset

(5

at;

by identigzigg along

IR+ ,

with order type

incompressible tori

gum

.

) that this class contains all Haken manifolds

with toral boundaries .

M.

Extend the _volume function to

not mrperbolic.

From

6.5.5

and

v(M)

6.5.2, we know

finite sum of volumes of hyperbolic manifolds.

sequence of values of v pieces of a manifold

Mi

. Express with

each

v(Mi)

=

wi wi

=

v3

-

H [M] ”0

when

that every value of

Suppose

{vi}

v

M

is

is a

is a bounded

as the sum of volumes of hyperbolic . The number of terms is bounded,

since there is a lower bound to the volume of a hyperbolic manifold, so we may

pass to an infinite subsequence where the number of terms in this expression is‘

constant. Since every infinite sequence of ordinals has an infinite non—decreasing subsequence, we may pass to a subsequence of

wi's

where all terms in these

6.28

expressions are non—decreasing. This proves that the set of values of v

well-ordered.

Furthermore,

our subsequence has a limit w

v“

+

is

...+vu

k

l

which is expressed as a sum of limits of non-decreasing sequences of volumes.

va is the volume Of a hyperbolic manifold M.j with at least as many cusps as the limiting number of cusps of the corresponding hyperbolic piece of Mi .

Each

Therefore, the

HE'S

may be glued together to obtain a manifold

This shows the set of values of v

mm

is closed.

M with v(M) = w

The fact that the order type is

can be deduced easily by showing that every values of v

isnot

derived set, for some integer k ; in fact, k :_ v / C , where

in the

kth

C is the

constant Just discussed.

6.7

Comensurability

6.7.1 m .

En

, then

isometries

r1

of

P1

If

and

F2

is oommensurahle with

En) to

6.7.2. Definition.

a group

Two

finited sheeted covers

Pi

if

such that

manifolds M1

El

and

Commensurability in either

P1 is conjugate (in the group of ri.n F2 has finite index in Pi

“2

and

M2

are commensurahle if they have

which are homeomorphic.

sense is an equivalence

relation, as the reader

easily verify.

6-7-3then

S3

are two discrete subgroups of isometries of

r2

and in

may

P2

If V

83 - W

—B

is the Whitehead link and

B is the Borromean rings,

has a four-sheeted cover homeomorphic with a two sheeted cover of

-

6-29

IV‘

I

liéiflb

83 w

33’s

V= 3106386 The homeomorphism involves cutting along a disk, twisting

Thus and

53 - W and S3 - B wl (S3 - B)

360°

are commensurable. One can see that

V: 7. 317731

and glueing back.

“l (S3 —

W)

are commensurable as discrete subgroups of

PSL (2 , m) by considering the tiling of

H3

by regular ideal octahedra.

Both

groups preserve this tiling, so they are contained in the full group of symmetries

of the octahedral tiling, with finite index. Therefore, they intersect each other with finite index.

Symmetries (octahedral tiling) n

(s3 _

:::

B) 1r

(s3-B)n

nl

(s3-W)

&

v1

CS3 - W)

6.30

Mas

I‘l

Two groups

and

r2

can be commensurable, and yet not be

conjugate to subgroups of finite index in a single group.

6.1.3. m . E Ml and

M2

is

Merbolic

m

i_sa

commensurable with

complete

M1

Merbolic manifold

, then

M2

is homotopy

with finite volume

eguivalent Eg‘a complete

manifold

This is a corollary of Mostow‘s theorem. Under the hypotheses,

finite

cover

M3

which is hyperbolic.

M2

M3

has a finite cover

Mh

M2

has a

which is a

"1(M1I») is a normal subgroup .of nl(MZ) . Consider the action of 111(M2‘) on “1(Mh) by conjugation. "1.(Mk) has a trivial center, so in other words the action of "1(Mh) on itself is effective. Then for every k . . . a c “1(M2) , Since some power a is in 1r1(M)+) , a must conjugate "1(Mh) non-trivially. Thus "1042) is isomorphic to a group of automorphisms of “1(Mh) , so by Mostow's theorem it is a discrete group of isometries of fin .

regular cover of

, so that

In the three dimensional case, it seems likely that hyperbolic. Waldhausen

[

Ml

would actually be

] proved that two Haken manifolds which are

homotopy equivalent are homeomorphic, so this would follow whenever

There are some sorts of properties of

NLl

is Haken.

3—manifolds which do not change under

passage to a finite—sheeted-cover. For this reason

(and for its own sake) it

would be interesting to have a better understanding of the commensura‘oility relation among

3—manifolds. This is difficult to approach from a purely topological point

of view, but there is a great deal of information about comensurability given by

a hyperbolic structure. For instance, in the case of a complete non-compact

6.31 hyperbolic

3-manifold M

of finite volume, each cusp gives

a

canonical Euclidean

structure on a torus, well-defined up to similarity. A convenient invariant for this

structure is obtained by'arranging M- so that the cusp is the point at

w

in the

upper half space model and one generator of the fundamental group of the cusp is a translation

2

l—r' z + l

0'1

of complex numbers

. A second generator

is then

2

|——>

z + a . The set

... ck corresponding to various cusps is an invariant of

the commensurability class of M

well defined up to the equivalence relation

nai+m G.

1

where

n,m,pq€Z

"I

pci+q

:1:

9

#0.(n,m,pandqdependoni)

(la

a

(¢+3) I

0

1L

ld+i

Modulus: “+3

6.32 In particular, if a

s

v»

, then they generate the same fields Inca) = 111(3) .

oi

Note that these invariants

6.1.h. W . g

H3 / I‘

are always algebraic numbers, in view of

1' i__§._ discrete suhEoup 5g PSL(2 , m)

has finite volume , then

such that

F _ii coangate _3 g gpoup _f matrices whose

entries are algebraic

Proof: This is another easy consequence of Mostow's theorem.

Conjugate

that some arbitrary element is a diagonal matrix

and some other

[3 0-u 1]

I‘ so

element is upper triangular ,

[0 xfl]

variety of representations of

I‘ having this form is 0 dimensional, by Mostow's

A

. The component

I' in the algebraic

of

theorem, so all entries are algebraic numbers.

One can ask,the more subtle question, whether all entries can be made algebraic integers.

Hyman Bass has proved the following

remarkable

result regarding this

question:

6.7.5 W volume where

Then

(Bass) . Let M

either

"in“

‘p_g conmlete

is. Cog-1ugate t_og

Mex-bolic 3—manifold g finite

9’ _iithe' ripg gt; algebraic integers , 9; M

surface (not homotopic

3’) . .a_ closed Mare—551m

subgroup 9; PSL(2 ,

contains

2.3.- cusp) . r

The proof is out of place here, so we omit it. See Bass,

[

J

. As an

example, very few knot complements seem to contain non-trivial closed incompressible

surfaces. The property that a finitely generated group subgroup of

PSL (2 ,

(9’)

I‘ is conjugate to a

is equivalent to the property that the

M group

6.33 of.matrices generated by

I is finitely generated. It is also equivalent to the

property that the trace of every element of easy to see

from

I is an algebraic integer. It is

this that every group commensurable with a subgroup of

is itself conjugate to a subgroup of PSL(2 , ‘algebraic integer, then an.eigenvalue

A ,

Hence

A-1

and Tr y

= A +

A-1

A

of

£9 ) . y

(If Tr

satisfies

yn

PSL(2 , 6})

= a is an

A2n — a An + l = O

.

are algebriac integers) .

If two manifolds are commensurable, then their volumes have a rational ratio.

We shall see examples in the next section of incommensurable manifolds with equal volume.

6.7.6

Questions . Does every

commensurability class of discrete subgroups of

PSL(2 , T) have a finite collection of maximal groups

(up to isomorphism)?

Is the set of volumes of 3—manifolds in a given commensurability class a discrete set, consisting of multiples of some number

Vo

?

6.8. _LSDBrawn—es.. Consider the

6.8.1

k—link chain

Ck' pictured below:

6.31;

If each link of the chain is spanned by a disk in the simplest way, the complement of the resulting complex is an open solid torus.

Y. S3

- Ck

c

is obtained from a solid torus, with the cell division below on its

boundary, by deleting the vertices and identifying.

it nx L\ BdDeC Y’Z’ lb

,

\

.A1

C

XYYZ Z

2

I

a:

0‘

C marl

C)‘ 0 'r’\

J

a

c

I

\

c

C

Z

IDUJ

BL

Y

E

6.35 To construct a hyperbolic structure for S3

Ck

, cut the solid torus into two

drums.

,ea.‘

Let P be a regular k-gon in of P

obtained by displacing

center, then P' height of

and P

H3

with all vertices on

SE2

If P'

is a copy

P along the perpendicular to P through its

can be joined to obtain a regular hyperbolic drum. The

P' must be adjusted so that the reflection through the diagonal of a

rectangular side of the drum is an isometry of the drum. If we subdivide the drum into

-J

2k pieces as shown,

6.36 the condition is that there are horospheres about the ideal vertices tangent to three faces. Placing the ideal vertex at

ain.upper half-space, we have

a

figure bounded by three vertical Euclidean planes and three Euclidean hemispheres

of equal radius

r .

an.

‘I\

n

Vi e w

79‘;m

a

)30 V6

From this figure, we can compute the dihedral angles a and be

c

= arc cos

,

B=r

§

Fn

v

(C3)

(commensurable with PSL (2 , Z /-_2)

below can also be given hyperbolic

structures obtained from a third kind of drum:

m

6.8.11

8

-.‘T

6.148

The regular drum is determined by its angles

a

and

B =w

-a

. Any pair

of angles works to give a hyperbolic structure; one verifies that when the angle

a

= arc cos

(cos

2%. —

l-)‘, 2

n = 1 gives a trivial knot. simplices with

on

F2

=

if

and we obtain

the

the hyperbolic structure is complete. The case

In the case n = 2 , the drums degenerate into

600 angles, and we .

When

F3

B

obtain

n = 3 , the

once more

angles

are

Passing to the limit

the hyperbolic structure

90° , the drums become octahedra

n =

a

, and dividing by Z}, we obtai:

following link, whose complement is commensurable with

6.8.12

O@

S3 if

v=

h.05977.. .

With these examples, many maps between link complements may be constructed. The

reader should experiment for himself.

One gets a feeling that volume is a very

good measure of the complexity of a link

is really inherent in 3-manifolds.

, and that the ordinal structure

7

Computation

g§_volume.

J. W. Milnor.

by

7.1.

The Lobachevsky function

.fl(e) .

This preliminary section will describe analytic properties, and

conJecture

number theoretic properties, for the function

JT(9)

=

-I

9 log

I2 sin ul

0

Compare Figure'Tdnl. Thus the first derivative and the second derivative

LObachevsg function

J("(e)

du .

is equal to

.QKB)

is equal to

- cot e

-

. I will call

log

[2 sin e

.fl(8)

the

(This name is not quite accurate historically, since

Lobachevsky's formulas for hyperbolic volume were expressed rather in terms of the

function

I for

|e| i_w/Z .

6 log (sec u) du

= J((e + n/2) + 9

2

0

11(6)

However our function

is clearly a close relative, and is

more convenient to work with in practice. Compare Clausen Another close relative of

_fl(e)

is the dilo arithm

2

a

w(z)

which has been studied by many authors.

[9] , [12] , [13]

log

Writing

w(z)

J

n 2 z /n

m

for

(See for example

2

log (1

- w)

[3])-

dw / w

I2]

3_ 1 ,

[1] , [2] , [8] ,

(where

MI

1

l) , the substitution w = log (1

- w)

e216

dw/w

(.1:

w(l)

-e

yields

- 29 + 21 log(.2 sin 9))

for 0 h both of these can be realized since 8 varies continuously from

0 to n-2 / n as the distance between the two base planes of

7.17

77a B

varies from

3

7.3.1

W:

0 to

There are an_infinite number g§_oriented three-manifolds whose

volume i_‘a_finite rational sum

7.h We

Thus we have the following:

gf_-r((6)

8's commensurable with v .

for

will now discuss an arithmetic method for constructing hyperbolic

3-manifolds with finite volume. The construction and computation of volume go back to Bianchi

and

Humbert. (See [5] , [7] , [10].) The idea is to consider

ring of integers in an imaginary quadratic field, Q

square free integer. Then PSL (2 ,

é}h)

(6:3), where

PSL (2 ,

£93)

PSL (2 , E) is the group of orientation preserving isometries of

W.

3;1

is a

is a discrete subgroup of PSL (2 , m)

Let P be a torsion free subgroup of finite index in

is an oriented hyperbolic three

d

6?; ,

- manifold.

H3

. Since

H3 / F

,

It always has finite volume.

Let Z [i] be the ring of Gaussian integers. A fundamental domain

for the action of PSL (2 , z [i] ) has finite volume. give different

manifolds; eg., there is a F

Different choices of

of index 12 such that

is diffeomorphic to the complement of the Whitehead link; another

2h leads to

the complement of the Borromean rings.

Incase d=3, there is a subgroup

r

(9’d

C: PSL (2

is ZIw]

where

, Z [w] ) of

index

.

H3 /

H3 / P of index

(N. Wielenberg, preprint). w

=

—l-2-—i

and

12 such that H3 /

diffeomorphic to the complement of the figure eight knot.

In order to calculate the volume of

r

PSL (2 ,

(R. Riley,

69d)

r

r

.

is

[11])-

in general we

the

7.18 recall the following definitions. Define the discriminant, Q

of the extension

Q- ( V -d ) to be

D

lid

9:1

If

a:

/

9d

Dedekind

CK (Y c

otherwise

II!

is considered as a lattice in

. The

- function

g

E

(S)

Q

runs through all ideals in

(S) is also equal to

7.h.l

s 3 (mod 1:)

d

for a field K

1

8’and

_

is defined to be

where

«(2)5

—1

II

f

VD / 2 is the area of

then

l

I 87W I denotes the norm of 07 taking all prime ideals N (a ) =

C

N(P)S

(Essentially due to Humbert) : Vol (H 3 / PSL (2 ,

(9’) d

>

T 33/2 2

t;

cur-E)

c

(2 ) /

m

This volume can be expressed in terms of Lobachevslq's function using Hecke's

C

(My/:5.)

ouadratic symbol

(i) If n = (ii)

(s)/:

01

(s)=

zc‘%)

n>0

/ns.

Here(%)isthe

where we use the conventions:

Pl

If’pID

, then

-

, Pt , pi

prime

(‘3)

o;

then

(“311)

11 ) ( ~13.

+1.

(

a p2 ) Pl ) L -2

(

*0'U

formula

7-19 (iii) For p

anodd prime

( ‘2 )

+1 if -D E X2 (mod p) = -1 if not

p

2

(iv) For p

'2 )

= +1 if -D = 1 (mod 8)

P

-1 if

#- 3

(-D

*

transform;

ie.,

(1)

X

(

.2 )

Multiplying by

-%

) is equal to

J15

e21rikn/D

l/n2

(

l /

V-D

times its Fourier

p n

and suming over

A 11'3“ -2 (2)2220 n

(

k

k mod D

8)

(mod h) by definition)

I——>

The function n

5 (mod

E

k=0

e 21rikn/D

:1

> O

we get

J—DZ()/n2 U

(

for some X

n>0

For fixed k the imaginary part of the left side is just the Fourier series for 2 JZ( "k / D ) . Since the right side is pure imaginary we have:

2

(3)

('E)JZ(_"k/D) X k mod D

Multiplying by

(h)

2—2

2

D /

kmodD

21+

Z(

n>0

2

)l/n

and using Hecke's formula leads to

('%)J(("k/D)=Vol(H3/PSL(2,(9d))

*Compare Hecke, Vorlesangen 'uber algebr. for help on this point.

Zahlen, p.

2111. I

am grateful to A. Adler

7.20 In the case 6. = 3 , (h) implies that the volume of is

4:: (1((1r/3)

the figure eight

12 in PSL (2 ,

- J‘((21r / 3)) knot S3 - K

% I((n/3)

) . Thus it has volume

. . volume computed by thinking of

S3

-K

as two

(PSL (2 , z [oj

. Recall that the complement of

H3 / r

is diffeomorphic to

z [a]

H3 /

where

6 J7 (1r/3) . .

copies

of

r

had index

This agrees with the

:w/3 , fl/3

, fl/3

tetrahedra glued together. Similarly the volumes for the complements of the Whitehead link and the Borromean rings can be computed using

in

(h) .

The answers

agree with those computed geometricall;

7.2. This algebraic construction also furnishes

an infinite number of hyperbolic

manifolds with volumes equal to rational, finite linear combinations of

J(

that any

rational

. Note

and

B would imply

relation between the volumes of these manifolds

could occur at

( a rational multiple of n)

that Conjectures A

most as a result of common factors of the integers, d , defining the quadratic fields.

In fact, quite

relations.

likely they would imply that there are no such rational

7.21 Reference 5

L. Euler, Institutiones calculi integralis, I, pp. 110

- 113

C. J. Hill, Journ, reins angew. Math. (CreL‘Le) 3 (1828), 101 T. Clausen, Journ. reine angew. Math. (Crelle)

I

(1768).

- 139.

8 (1932), 298

- 300V.

I

N. Lobachevshy, Imaginary Geometry and its Application to Integration (Russian) Kasan

1836. German translation, Leipzig 1904-.

"13. L

Laptev, Kazan Gos Univ.

Bianchi, Math. Ann. Gieseking, Thesis,

38

U6.

(1891) 313

Munster

111. (19511), 53

- 77.)

- 333 and 1+0 (1892), 332 - 1:12.

1912. (See Magnus, Noneuclidean

and their Groups, Acad. Press

. Humbert, Comptes Rendus

Zapiski

(For a modern presentation see

1971+,

Tesselations

p. 153.)

169 (1919) M48

- 1:511.

S. M. Coxeter, Quarterly J. Math. 6 (1935), 13

- 29.

Lewin, Dilogarithms and Associated Functions, Macdonald (Laldon),

G. Swan, Bull. AMS

7h (1968), 576

Riley, Proc. Cambr.

Phil.

Soc.

1958.

- 581.

77 (1975) 281

- 288.

M- Gabrielov, I. M. Gel'fand, and M. V Losik, Fmetional Anal. and Appl. 9

IJ

.

(1975), pp- h9 , 193.

13.

. Bloch, to appear.

111». D. Kubert

and S. Lang, to appear.

8.1 Kleinian Groups

Our discussion so far has centered on hyperbolic manifolds which are closed, or at least complete with

finite volume.

The theory of

complete

hyperbolic manifolds with infinite volume takes on a somewhat different

character. Such manifolds occur very naturally as covering spaces of ClOSEd manifolds.

They also arise in the study of hyperbolic structures on

3~manifolds whose boundary has

compact

negative Euler

characteristic. We

will study such manifolds by passing back and forth between the manifold and

the action of its fundamental group on the disk.

8-1- Lhe Let

H11

sea.

I_imm

r

be any discrete group of orientation-preserving isometries of

. If x e

an

L1. c. sfl'l is defined to be Ix of x . One readily sees

is any point, the limit set

the set of accumulation points of the orbit

that L model.

I

is independent of the choice of x

If y

elements of

r

€ZHn

is any other point and if

O ; hence

The group

{Yix} d(yix , yiy)

such that

hyperbolic distance goes to

r

by picturing the Poincaré disk

lim

{vi}

is a sequence of

converges to a point on

82-1 ,

the

is constant so the Euclidean distance

yiy = lim yix .

is called elementg if the limit set consists of

O ,l

or 2 points.

8.1.1.

W. r

finite index.

—I

i_s elementfl iff

r

hasg abelian subgzoup g

8.2

When

I‘ is not elementary, then

LI‘

is also the limit set of any orbit

the

sphere at infinity. Another way to put it is this:

8.1.2.

. I; I' _i£_n0t elementary , then evgy

on

subset

31: So

M.

invariant by I‘. contains

Let K: S... be any closed

elementary,

set

LI‘

closed

.

invariant by

I‘ . Since I‘ is not

K contains more than one element. Consider the projective

(Klein) model for

H11

, and let H(K) denote the convex hull of K .

H(K) may be regarded either as the Euclidean convex bull, or equivalently, as the hyperbolic convex hull in the sense that it is the intersection of

"intersection" with

all hyperbolic half-spaces whose Clearly

H(K)n

s“

Sun

contains K .

= K.

H(K)

Hem Since K

is invariantby

point in

Hun

r ,

H(K)

is also invariant by

H(K) , the limit set of the orbit

the closed set H(K) . Therefore

LI‘ C K

.

1'x

MOJCI Ior H,5 r.

If x

is any

must be contained in

8.3 A closed set K invariant by a group T which contains no smaller closed invariant set is called a minimal set. It is easy to show, by Zorn's

lemma, that a closed invariant set always contains at least one minimal set.

It is remarkable that in the present situation,

8.1.3. W. If §_normal

suonup, then

I‘ i_sa non—elementm

LF'

=

LP

wand

,laér'dr

En which would be that Lr! 2’

Examples

If

M2

invariant by

LT

i_s

P' to itself, hence it takes

r' must be infinite, otherwise F' would have a

to L1"!

set for F

.

Proof. An element of I conjugates

8.1.2

LP is the unique minimal

P

so

Lr,

fixed point in

I would be finite. It follows from

. The opposite inclusion is immediate.

is a hyperbolic surface, we may regard

of isometries of a hyperbolic plane in

H3 .

nl(M)

as a group

The limit set is a circle. A

group with limit set contained in a geometric circle is called a Fuchsian

group

a

The limit set for a closed hyperbolic manifold is the entire sphere

If

M3

is a closed hyperbolic

82-1 .

3-manifold which fibers over the circle,

then the fundamental group of the fiber is a normal subgroup, hence its limit

set is the entire sphere. For instance, the figure eight knot complement has

fundamental group

< A,B :

ABA-J'BA = RAB-1A3 >

0A"8A=BA

It fibers over group

«1(F)

S1

with fiber F a punctured torus. The fundamental

is the commutator subgroup, generated by

AJB-l

A-lB . 82 even

and

Thus, the limit set of a finitely generated group may be all of when the quotient space does not have finite volume.

A more typical example of a free group action is a Schottky group, whose limit set is a Cantor set. considering

Hn

minus

Examples of Schottky groups may be obtained by

2k disjoint half-spaces, bounded by

hyperplanes:

If

we choose isometric identifications between pairs of the bounding hyperplanes,

we obtain a complete hyperbolic manifold with fundamental group the free group on k

generators.

Sc.hott I“) SYOUP n

2

8.5 It is easy to see that the limit set for the group of covering transformations is a Cantor set.

8.2.

The domain 9; discontinuity The domain of discontinuity for a discrete group

Dr

=

89n-l - Lr

P is defined to be

. A discrete subgroup of PSL(2,T) whose domain of

discontinuity is non-empty is called a ICLeinian

gpoup.

(There are actually

two ways in which the term Kleinian group is generally used. Some people refer to any discrete subgroup of PSL(2,E) distinguish between a type

group, where DP

# ¢.

as a Kleinian group, and then

I group, for which

LP

=

Si ,

and a type II

As a field of mathematics, it makes sense for

Kleinian groups to cover both cases, but as mathematical objects it seems

Dr

useful to have a word to distinguish between these cases

# ¢

and

Dr=¢') We have seen that the action of P on

as much as

LP

is minimal

In contrast, the action of r on

possible.

Dr

- it mixes up LT is as discrete

as possible.

8.2.1.

.

If

r

is a group acting on a locally compact

space X ,

the action is properly discontinuous if for every compact set K C‘X , there

are only finitely many

7 6.? such that

7K 0 K ¢ ¢ .

Another way to put this is to say that for any compact set K , the map

F

X

K -* X given by the action is a proper map, where T has the discrete

(Otherwise there would be a compact set K'

topology. preimage of

K'

would carry K

is nonrcompact.

compact

Then infinitely many elements of

I2_ I acts pronerly discontinucualy pp the locally

I;

Hausdorff space X , then the auotient space X i§_Hausdorff.

x—> x/r i_sg coveripg projection

the action i_sfree, the quotient man

x1 , x2 6 X

Proof. Let

be points on distinct orbits of

Nl be a compact neighborhood of xl , so we may assume intersect

(J

7

yer

‘

N1

Nl

Then

r

U K' to itself) .

mm .

8.2.2

such that the

N1

P . Let

Finitely many translates of

is disjoint from the orbit of

gives an invariant neighborhood of

X1

x2

x2

.

disjoint from

N1

x2 . Similarly, x2 has an invariant neighborhood N2 disjoint from ; this shows that X / r is Hausdorff. If the action of r is free, we may find, again by a similar argument, a neighborhood of any point disjoint

to X /

from

r .

x which is

all its translates. This neighborhood projects homeomorphically

r

.

Since

acts transitively on the sheets of X

it is immediate that the

projection

X -+ X /

r

over X / F ,

is an even covering, hence

a covering space.

gpoung M g H“ , the and ip fact pp_ Hn U Dr) i§_properly discontinuous action Eran]?r .——

8.2.3. Wan .

Proof.

ball

to

r

i g a_sm

convex hull H(Lr). There H(Lr) defined as follows.

Consider the

Hnu s”

If

is a retraction r

of the

8.7 If x 6

H(LT)

H(LI')

. If x

, r(x) = x . Otherwise, map is an infinite point

to be the first point of touches

because

Lr

H(Lr)

in

Dr

.1:

to the nearest point of

, the nearest points is interpreted

where a horosphere

"centered" about x

. This point r(x) is always uniquely defined

H(Lr)

is convex, and spheres or horospheres about a point in the

ball are strictly convex. Clearly r is a proper map of

H(Lr)

- LI.

. The action of r on

discontinuous, since property

for

Hn'U Dr

r

H(LI.)

- LI,

Hn U Dr

to

is obviously properly

is a discrete group of isometries of

H(Lr)-

LT

follows immediately.

Remark. This proof doesn't work for certain elementary groups; we will ignore such technicalities.

It is both easy and common to confuse the definition of properly

discontinuous with other similar properties. To give two examples, one might make these

definitions:

; the

8.8 8.2.h. neighborhood N

. The action of I‘ is

wandering if every point has a

such that only finitely many translates of N

intersect

N.

8.2.5. Psalm.

The action of

I‘ has discrete orbits if every orbit of

I' has an empty limit set.

IE

X, the projection

—* X /

>4

8.2.6. Proposition.

rgafrre,uandanin¢action9_na

.

W

space

1‘. is a miss mismana-

Proof. An exercise. Warning

Even when X

is a manifold, X / I‘ may not be Hausdorff.

For instance, consider the map

L:

IR2-0—.> 132-0

L (any)

)

It is easy to see this is a wandering action. The quotient space is a surface with fundamental group Z 0 Z

The surface is non-Hausdorff, however,

8.9 since points such as (1,0) and (0,1)

do not have disjoint neighborhoods.

Such examples arise commonly and naturally; it is wise to be aware of this phenomenon.

r

The property that pair of points_ x , y

disjoint from y

of

Kleinian

.

groups

has discrete orbits simply means that for every

in the quotient space X /

r

, x has a neighborhood

This can occur, for instance, in-a l-parameter family

I‘t

9

t 5 [0,1]

.

There are examples where

and the family defines an action of Z: on orbits which is not a wandering action. See

[O , l] x §

33

I‘t

= Z ,

with discrete

. It is remarkable

that the action of a Kleinian group on the set of all points with discrete

orbits is properly discontinuous.

8.3.

Convex

huerbolic

manifolds

The limit set of a group action is determined by a limiting process,

so that it is often hard to "know" the limit set directly. The condition that a given group action is discrete involves infinitely many group elements,

so it is difficult to verify directly. Thus it is important to have a

concrete object, satisfying concrete conditions, corresponding to a discrete group action.

We consider for the present only groups acting freely.

8.3.1. convex

. A complete hyperbolic manifold M with boundary is if every path in M

is homotopic (rel endpoints) to a geodesic arc.

8.10 (The degenerate case of an arc which is a single point may occur).

8.3.2. We.. developing map D : Proof.

If

M

A 2L: iete hyperbolic manifold M _ii convex iff the _—

M —> En

i_s a homeomorphism Q _a_. convex subset g

is a convex subset

S of

En

H11

.

, then it is clear that

M is convex, since any path in M lifts to a path in S , which is homotopic to a geodesic arc in S , hence in M .

If' M

is convex, then D

is

l

-l,

since any two points in

may be joined by a path, which is homotopic in M geodesic arc.

MM)

and hence in

M

M

to a

D must take the endpoints of a geodesic arc to distinct points.

is clearly convex.

We need also a local criterion for M to be convex. We can define M to be locally convex if each point

aM

CoY‘

he A

x,

bar ’1". »

x e: M has a neighborhood isometric to a convex subset of

DOA

En

. If x e: M ,

then x will be on the boundary of this set. It is easy to convince oneself

that local convexity implies conVexity: picture a path, and imagine

8.11 straightening it out. Because of local convexity, one never needs to push it out of

8M . To make this a rigorous argument, given a path p of length

2 there is an e

CZ7L (p0)

such that any path of length i_s

is homotopic to a geodesic arc.

of length between

a / h

intervals in turn, putting

s / 2 .

and

intersecting

Subdivide p

Straighten out

a new division point

into subintervals

adjacent pairs of

in the middle of the resulting

arc unless it has length §_s/2 . Any time an interval becomes too small, change the subdivision. This process converges, giving a homotopy of p

to a geodesic

arc,

since any time there are angles

not close to v ,

the homotopy significantly shortens the path.

Local convexity convexity

M This gives us a very concrete object corresponding to a Kleinian group:

3—manifold M with non-empty boundary.

a complete convex hyperbolic Given a. convex manifold

M , we can define H(M) to be the intersection

of all convex submanifolds M'

of M

such that

n1 M' -—+ v1 M

is an

8.12

isomorphism. H(M)

is clearly the same as

H

Lnl(M)/ «1(M)

. H(M) is a

convex manifold, with the same dimension as M except in degenerate cases.

8.3.3. W . EL small

I_f; M "_sg compact convex

deformation

511 htl to give _E_I

marbolio

g the Methane structure 23

a new

M

manifold , then

can

__

‘33 enlar

ed

erbolic manifold __._1'P_ homeomo hic t_o M .

convex

Proof. A convex manifold is strictly convex if every geodesic arc in

M has interior in the interior of M . If M is not already strictly convex, it can be enlarged slightly to make it strictly convex.

follows from the fact that a neighborhood of radius

a

(This

about a hyperplane

is strictly convex) .

le convex

Stric

Convex

Thus we may assume that M'

is a hyperbolic structure that is a slight

defamation of a strictly convex manifold M . We may assume that our deformation M'

manifold length

M"

is small enough that it can be enlarged to a hyperbolic

which contains a

2. greater than

Ze-neighborhood of M' .

a in M has the middle

(9.

Every arc

- a)

of

some uniform

8.13 5 from 3M ; we may take our deformation M' of M small enough

distance

that such intervals in M' of

M'.

have the middle

2.

-a

still in the interior

This implies that the union of the convex hulls of intersections Of balls

of radius

36

IVith‘M'

is locally convex,

hence convex.

The convex hull of a uniformly small deformation of a uniformly convex manifold is locally determined.

When M

Remark

8.3.3

is non-compact, the proof of

applies provided that

M has a uniformly convex neighborhood and we consider only uniformly small deformations. We will study deformations in more generality in

8.3.h.

W.

hyperbolic manifolds

M: gag M: g M: Ed suppose .45 I“? Suppose

homeomoghism f :

wllll1

t_o

1'le .

strictly convex, compact

:

Liv-ans diffeomorphism 93

EBn —* En

§

—>

K

_i_s_g homotopy euuivalence

3M1 - Tar-amass quasi-conform o_f_ E Poincaré disk to itself con

fi_sa pseudo-isometry

o_an-

ati

8.1h

Proof.

Let m

¢

assume that of

M1

$

_

two normal rays

yp

l

p2

d

or

t

from

.

yp

2

8M2 3M1 ,

has a unique normal

let

is the distance and

6 is the angle between the normals of p1 and

From this it is evident that f is a pseudo-isometry extending of isometries of

r

8.3-5.

En

3'.

, there are

(which are manifolds

at least four distinct and interesting quotient spaces

acts freely). Let us name them:

Definition.

MT Nr

H(Lr)

.

‘.

T

(3“ u

Pr

(ant; Dr

or

the convex hull quotient

, the complete hyperbolic manifold without boundary

Dr)

/

r ,

u wr,\

the Kleinian manifold

wr

the t :1 . Here se is 1 dual to planes in of points in the projective mode

H(NI‘)

/

r

whose intersection with

=

MT C Nrc or c PT

acts properly discontinuously on

discontinuity on

and

r ,

or

We have incluSions fact that

/

En / r

En

Mr

We may

at nearby points is approximately cosh t + a sinh t ,

Associated with a discrete group F

when

M2

f(x) be the point on Y% . ¢(p, The distance between points at a distance.of t along

8M2

and

3M1

on

p

has distance

-

_——v—

where

.

Each point

ray 7 ; if x E.y p p a distance t from

to

a pseudo-isometry between the developing images

13 already

M2 .

and

Ml

¢ to a map from

be a lift of

En U Dr

.

MT

,

NT

and

.

S“ It

.

is

is contained in .

easy to derive

En U Dr U WT

Or

Dr

t he

from the proper

have the same homotopy type.

are homeomorphic except in degenerate

cases,

and

Nr

int

(Or)

8.15

PP

LP

is not always connected when

is not connected.

8.h. Geometrically finite gpoun .

r

8.h.l. Definition.

{’3gite if

is eom t

“Va (Mr)

has finite

volume.

‘715 (Mr)

The reason that

MT

isirequired to have finite volume,and not

, is to rule out the case that P is an arbitrary discrete group of

isometries of

Hn-lC: En

. We shall soon prove that geometrically finite

means geometrically finite (8.h.3

8.h.2.

mm .

has full measure

r

just

on

SQ

E

) .

I_‘ i_s a__xeometric&ll finite, then

LI.

3 o measure . I_f

L1. c s,

has full measure , the action

g

is ergodic.

Proof. This statement is equivalent to the assertion that every bounded

measurable function f

a.e.

L

supported on

T

and invariant by

(with respect to Lebesque measure on

consider the function

the points on

SQ

"visual" measure

on

En

determined by

Following Ahlfors, we

f as follows. If x e

En

correspond to rays through x ; these rays have a natural

Vx

to the visual measure gradient flow of

hf

so).

F is constant

hf

. Define

Vx

hf(x)

to be the average of f with respect

. This function

hf

is harmonic, i.e., the

preserves volume, div grad

For this reason, the measure

hf

Vx

= 0 is called harmonic measure.

,

8.16 To prove this, consider the contribution to area A

e.sn'l

centered at p

hf

coming from an infinitesimal

. (i.e., a Green's function). As x moves

a distance d in the direction of p , the visual measure of A in proportion to function of A

to

e(n-l)d

.

The gradient of any multiple of the characteristic

is in the direction of

e(n—l)d . The

goes up exponentially,

p , and also proportional in size

flow lines of the gradient are orthogonal trajectories

to horospheres; this flow contracts linear dimensions along the horosphere in

e.(1 , so

proportion to

it preserves volume.

A

.-

/

H(Lr)

be the canonical retraction. If G, is any fundamental polyhedron for the action of Z in some neighborhood of p in

H(Lr)

then

r-1(Q,)

is a

fundamental polyhedron

6\\\£

.

EB U Dr

in some neighborhood of p

21

.

A fundamental polyhedron near the cusps is easily extended to a global

fundamental polyhedron, since (c) '9 (a)

way

Suppose that

point x|e P n S.

A of P

.

n

x

S.

Or -

(neighborhoods of the cusps)

is compact.

T has a finiteysided fundamental polyhedron P .

is a regular point

(6

Dr)

if it is in the interior

or of some finite union of translates of P . Thus, the only

can be a limit point is for x to be

a point of tangency of sides

of infinitely many translates of P . Since P

points of tangency of sides, infinitely-many y

to x, so x is a fixed point for some element 7 the translates of P by powers of

arranged to be

a

in upper

7 would

half-space,

can have only finitely many

I must

r.

identify one of these points

7 must be parabolic, otherwise

limit on the axis of y . If x it is easy to see that

LT

E

must

is

be

contained in a strip of finite width: (Finitely many translates of P must form a fundamental domain for

{1n}, acting

finite index in the group fixing

-

on some horoball centered at

9, since

{yn}

has

. The faces of these translates of P which do notpass

8.22

through

0

T outside this finite

lie on hemispheres. Every point in

{yn}

collection of hemispheres and their translates by

It follows that v

('776 (M)) = v(‘77e IT‘A

the contribution near any point of

’77 s (H (Lr)) n P 8.5.

(H

lies in

(LT)) n P)

Dr

)

is finite, since

P is finite, and the rest of

is compact.

The geometgz of

the hounggy

Consider a closed curve

of the convex hull.

in Euclidean space, and its convex hull H(o)

a

.

The boundary of a convex body always has non-negative Gaussian curvature. 0n the other hand, each point

in

p

33(0)

-

line segment or triangle with vertices on

segment on

3H(o) through

at p . It follows that

p

3H(o)

0

o . Thus, there is some line

, so that

-

lies in the interior of some

33(0) has non-positive curvature

c has zero curvature, i.e., it is

"developable". If you are not familiar with this

idea, you can see it by

bending a curve out of a piece of stiff wire (like

coathanger). Now roll

the wire around on a big piece of paper, tracing out a curve where the wire

touches. Sometimes, the wire may touch at three or more points; this gives

alternate ways to roll, and you should carefully follow all of them. Cut out the region in the plane bounded by this curve (piecing if necessary).

By

taping the paper together, you can envelope the wire in a nice paper model of its convex hull.

The physical process of unrolling a develOpable surface

onto the plane is the origin of the notion of the developing map.

8.23

The

same

physical notion applies in hyperbolic

any closed set on

Sulu

, then H(K)

lies on a line segment in

hyperbolic plane.

curvature

3-space. If K is

is convex, yet each point on

8H(K) . Thus, 3H(K)

3H(K)

can be developed to a

(In terms of Riemannian geometry,

3H(K) has extrinsic

O , so its intrinsic curvature is the ambient sectional curvature,

-l . Note however that

BH(K)

is not usually differentiable) . Thus

BH(K)

has the natural structure of a complete hyperbolic surface.

8.5.1. i_s a

r o

.

‘t'o . If T

hflerbolic

EL torsion

free fleiniau group,

then

3M?

surface .

MT is of course not generally flat - it is bent in y c:aMT consist of those points which are not in the

The boundary of

some pattern.

is

Let

8.21;

3MT . Through each point x in y , there 5x is also a geodesic in the hyperbolic is a. unique geodesic 8x on 8M1. . structure of BMI, . y is a closed set. If 3M1. has finite area, then Y is compact, since a neighborhood of each cusp of 3M1. is flat. (See §8.h). interior of a flat region of

8.5.2. A CM

Definition. A lamination L

on a manifold

M”1

(the support of L) with a local product structure for A . More

_ IRn 1‘

precisely, there is a covering of a neighborhood of A

Ui

neighborhoods

JED-k the form ¢iJ form

is a closed subset

x

¢~

—1+

B , B

CIRk (fij

(x , y)

x

131‘

oi

so that

in M with coordinate

(A n

. The coordinate changes (x , y) ,

giJ (y)

)

when

U1) is of the (#13 must be of y éB . A lamination

is like a foliation of a closed subset of M . Leaves of the lamination are

defined just as for a foliation.

males .

If

98

is a foliation of M

closure of the union of leaves which meet

Any submanifold of a manifold M Clearly, the bending locus

whenever two points of

y

y

and S c M

is any set, the

S is a lamination.

is a lamination, with a single leaf.

for

Mr

has the structure of a lamination:

are nearby, the directions of bending must be

nearly parallel in order that the lines of bending do not intersect. A

lamination whose leaves are geodesics we will call a geodesic lamination

8.25

a‘—-' \ dm ’1 qe

odeSiC

Maxim).(on

By consideration of Euler characteristic, the lamination 7 cannot hava all of

3M as its support, or in other words it cannot be a foliation. The

complement

3M

-

y

consists of regions bounded by closed geodesics and

infinite geodesics. Each of these regions can be doubled along its boundary to give a complete hyperbolic surface, which of course has finite area. There

-

_v-

p’

8.26

is a lower bound of n for the area of such 2

I 3 (8M) I

lamination y

a region, hence an upper bound of

for the number of components of

3M

-

y

.

Every geodesic

on a hyperbolic surface S can be extended to a foliation with

isolated singularities on the complement. There

is an index formula

for

the Euler characteristic of

S in terms of these

\

singularities. Here are some values for the index.

UM

8.27 From the existence of an index formula, one concludes that the Euler S is half the Euler characteristic of the double of

characteristic of

S

-

. By the Gauss-Bonnet theorem,

y

Area (S or in other words, y

— y)

has measure

= Area (S) 0

To give an idea of the range of possibilities for geodesic laminations,

{Yi}

one can consider an arbitrary sequence

of geodesiclaminations:

[Yi] converges

simple closed curves, for instance.. Let us say that

to Y

if for each x 6 support y , and for each

enough

‘77:

Yi n x

the support of

i

. Note

(x) are within

a

Yi

intersects

a ,

g776(x)

of the direction of

the

geometrically

for all great

and the leaves of

leaf of y

through

that the support of y may be smaller than the limiting support of

so the limit of a sequence may not be unique. See argument shows that every

sequenge {vi}

58.10. An

Yi

easy diagonal

has a subsequence which converges

geometrically. From limits of sequences of simple closed geodesics, uncountably

many geodesic laminations are obtained.

Geodesic laminations on two homeomorphic hyperbolic surfaces may compared by passing to the circle at

by a pair of points diagonal

l (Su'

x

{(x

See1

, x)} . A

-A)

Topologically,

(x1 , x2

/ Z2

)

0 b_e

arbitrg . There i_s some a > 0 such that for all geodesic laminations

Y 3; S , the train track approximation such 5 way that all branch lines g

Te

Proof. Note first that by making the leaves of the

foliation

J

there would be a sequence of containing an open set.

When all branches of

Ta

1e_— can be

are

a

C2

realizedgz; S

curves with curvature < 6 .

sufficiently small, one can make

very short, uniformly for all Y

Y's

E

: otherwise

converging to a geodesic lamination

[One can also see this directly from

area considerations]-

are reasonably long, one can simply choose the tangent

vectors to the switches to be tangent to any geodesic of Y where it crosses the corresponding leaf of

j

; the

branches can be filled in

by curves of

small curvature. When some of the branch lines are short, group each set of

.

switches connected by very short branch lines together. First map each of

these sets into S

8.9.3. W

-

,

then extend over the reasonably long branches.

Messiaen Lamination ahishiscgnisabxaclosa

train track approximation

To

23 a

geodesic lamination Y

has

all leaves

close t_o M pf Y . Proof:

This follows from the elementary geometrical fact that a curve in

hyperbolic space with uniformly small curvature is uniformly close to a unique geodesic.

(One way to see this is by considering the planes perpendicular to r

8.55 the curve

-

they always advance at a uniform rate, so in particular the

curve crosses each one only once.)

—

8.9.h. W . A

lamination A

lamination iff

g;surface s i_s isotopic t_g seedesic

i_s carried.flwflafl

(a) A

1

ad“) 1km we SikfikaEIm-EE(g—inrinite)EQT-

2322:.

Given an arbitrary train track' T , it is easy to construct pep;

hyperbolic sturcture for

S on which

curvature. The leaves of A near

1

T

is realized by lines with small

then correspond to a set of geodesics on

. These geodesics do not cross, since the

Condition

(b) means that distinct leaves of A

When leaves of

A

leaves of

A

S ,

do not.

determine distinct geodesics.

are close, they must follow the same path for a long finite

interval, which implies the corresponding geodesics are close. Thus, we obtain a geodesic lamination Y

which is isotopic to

A . (To have an isotopy,

it suffices to construct a homeomorphism homotopic to the identity.

homeomorphism is constructed first in a neighborhood of

T

This

, then on the rest

of S) .

Remark . From this, one sees that as the hyperbolic structure on S varies, the corresponding geodesic laminations are all isotopic. This issue was quietly skirted in

When

58.5.

a lamination

associate a number

A has an invariant measure u , this gives a way to

h(b) to each branch line b of any train track which

8.56 dominates Y : h(b)

is just the

transverse measure of

the leaves of

A

collapsed to a point on b . At a switch , the sum of the "entering" numbers equals the sum of the "exiting" numbers.

,7

/.:>\

1.8 6

L1 5‘

2' 0 Conversely, any assignment of numbers satisfying the switch condition determines

a unique geodesic lamination with transverse measure: line b

of

foliation

g

1

to a corridor of constant width by equally spaced lines.

first widen each branch

u(b) , and give it a

3

3'

Z

8.57 As in 8.9.h,this determines a lamination Y ; possibly there are.many leaves of

the

19

collapsed to a single leaf of Y , if these leaves of

same

infinite path.

.88

4?

all have

has a transverse measure, defined by the distance

between leaves; this goes over to a transverse measure for Y .

8.10.

3-manifolds

Realizing laminations in

For a quasi-Fuchsian group P , it was relatively easy to "realize" a

corresponding surface in MT

geodesic lamination of the

at infinity.

, by using the circle

However, not every complete hyperbolic 3—manifold whose

fundamental group is isomorphic to a surface group is quasi-Fuchsian, so we must make a more systematic study of realizability of geodesic laminations.

8.10.1.

Definition.

Let f : S ——+ N be a map of a hyperbolic surface to

a hyperbolic 3-manifold which sends cusps to cusps. A geodesic lamination Y is realizable in the homotopy class of

a cusp-preserving homotopy)

8.10.2. I

. . . I;_

f if

f

is homotopic

(by

to a map.sending each leaf of Y to a geodesic.

ii realizable ip_the

Y

the realization ‘;§ (essentially ) unique: that

pp;

class

homoto

the

2;, f ,

p§_each leaf 2;

LEI

on S

Y i§_uni uel determined Proof.

Consider a lift

N . If S is closed,

h

3

of a homotopy connecting two maps of

S into

moves every point a bounded distance, so it can't

move a geodesic to a different geodesic. LIf S has cusps, the homotopy can be modified near the cusps of S

so

8

again is bounded.

8.58 In section 8.5, we touched on the notion of The geometric topolog

geodesic laminations.

geometric

on

convergence of

geodesic laminations is

the topology of geometric convergence, that is, a neighborhood of Y of laminations

Y'

which have leaves near every point of

parallel to the leaves of Y . If

curves, then neighborhood

fix

. The

track approximations of Y

The measure topolog on

(of full support)

Y2

are disjoint simple closed

(Y , 11)

geodesic laminations compatible with a

Y .

give a neighborhood basis for

SEOdF—‘Sie

laminations with transverse measures

is the topology induced from the weak topolow on measures

in the Moebius band J of

and

Y , and nearly

Y1 U Y2 is in every neighborhood of Y1 as well as in every of Y2 . The space of geodesic laminations on S with the geometric

topology we shall denote train

Y1

consists

outside

consists of

S”

in the Klein model. That is, a neighborhood

(Y' , 11') such that for a finite set

of continuous functions with compact support in J

,

I

I fi du -

fl

, ..., fk

J fi

du'


O is a real number.

8.10.3. W . o_n;_thl m_a_.p_

772$

Let

be the space of

——~M

modulo

c

8.59 8.1o.h.

W . ’Themm

2each uncrumpled Proof g

W

:

W

#1585

(s , N)

(s) which

a_ss;525

surface its wrinklig locus i_s continuous .

8.10.3.

For any point x in the support of a measure

11

U of x , the support of a measure close enough to u

and any neighborhood

must intersect U .

_Pro_of g 8.1o.h. An interval which is bent Cannot suddenly straightenAway from any cusps, there is a positive infimum to the an interval of length

"amount" of

bending of

a which intersects the wrinkling locus w(S)

middle third, and makes an angle of at least

in its

"amount"

a with w(S) . (The

of bending can be measured, say, by the difference between the length of

a

and the distance between the image endpoints) . All such arcs must still cross

w(S')

for any nearby uncrumpled surface S' .

When

I

S has cusps, we are also interested in measures supported on compact

geodesic lamination.

This space we denote

track diescription for

(Y , u) ,

then neighborhoods for

(Y

|u(b) - u'(b)|

< a

9

m

777560(S). h(b)

75

('r , u)

If

is a train

0 fipflbranch of

u) are described by {(‘r' , u')}, where

. (If b is a branch of r' not in

'r

branches of

1:

,

1" and

, u(b) = O by definitic:

In fact, one can always choose a hyperbolic structure on S so that good approximation

1' c

1:

T

is a

to Y . If S is closed, it is always possible to squeeze

together along non-trivial arcs in the complementary regions to

obtain a new train track which cannot be enlarged.

8.60

This implies that a. neighborhood of

number of real parameters.

W: (S)

has cusps,

212:.boundfl

M.

7774?

4335

(s) i_s coppact ,9;

a0flow) m

compact branch of

Te

is compact.

5

Similarly, when

”560(8) 10$ (s)

S

i_s_g compact

i_s at. compact

Tl ,..., 1k

carrying every

(There is an upper bound to 'the length of a

, when- S and e are fixed). The set of projective

classes of measures on any particular

remarks. In

s

There is a finite set of train tracks

possible geodesic lamination.

P x (S)

(S) is a. manifold.

is a. manifold with boundary

8.10.5. W33. manifold

Thus,

(Y , u) is parametrized by a. finite

That

9d“: (S)

1:

is obviously compact, so this implies

is a. manifold

follows from the preceding

we shall see that in fact it is the simplest of possible

manifolds .

In

8.5

, we indicated one proof of the compactness of

”at

(S) . Another

8.61 proof goes as follows.

First, note that

8.10.6. W . Every measure

1.:

, (possibly with smaller support).

Br_oof_£ 8.10.6. Choose leaves of V

(33's

such that the total number

goes to infinity. Let

times the counting measure of converging

a1

a finite set of transversals

meet every leaf of y . Suppose there is a sequence

the

admits some transverse

geodesic lamination y

2i

“i n

Ni

{1i}

. The sequence

(in the weak topology) to _a measure

which

of intervals on

of intersections of

be the measure on

aJ

,..., “k

Uov.J

{1.11}

Li

with

which is l /

Ni

has a subsequence

u . 'It is easy to see

that

u

is invariant under local projections along leaves of y , so that it determines

a' transverse measure. If there is no such sequence counting

every leaf is proper, so the

measure for any leaf will do.

Continuation fiproof g

'0) cf

{2i} then

(S) in

iffi (S)

8.10.5:

compact

intersects the closure of every point of

set

‘1‘

;

g

4‘

of

Because of this fact, the image

Any collection of open sets which covers which covers the

3-10-5

J35

22556

(S)

(S) has a finite subcollection

therefore, it covers all of

flat

(S)

Armed with topology, we return to the question of realizing geodesic lamination

Let

R f C Joe—(S)

consist of the laminations realizable in the homotopy

class of f First, if y

consists of finitely many simple closed geodesics, then

y

8.62

is realizable provided

nl(f)

maps each of these simple closed curves to

non—trivial, non-parabolic elements.

If we add finitely many geodesics whose ends spiral around these closed geodesics or converge toward cusps the resulting lamination is also realizable

except in the degenerate case that f restricted to an appropriate non-trivial pair of pants on

S factors through a map to

S1 .

To see this, consider for instance the case of a geodesic g on S whose

f Ito H3 , we see that the two ends of f(z) are asymptotic to geodesics ?(El) and ¥(32) Then f is homotopic to a map taking g to a geodesic unless ¥(§l) and ?(EZ) converge to the same point on S“ , which can only happen if ¥geodesic lamination

(Logically, one can of

M2 into H3

v on S

is realizable in

near 9 , every

N' near its realization in N .

, with the compact—open topology, so that

Mi

9'

think of uncrumpled surfaces as equivariant uncrumpled maps

Choose any subsequence of the components of

___

converges strongl‘;

converge

pi's

injflfifs)

"nearness" makes

sense).

so that the bending loci for the two boundary

. Then the two boundary components must

converge to locally convex disjoint embeddings of S in N (unless the limit is Fuchsian).

M

These two surfaces are homotopic, hence they bound a convex submanifold

N , so p(I)

of

Since

8.3.3:

M[s

a)

is geometrically finite. is compact, strong convergence of

{pi}

follows from

3

no unexpected

perturbation of

identifications

N can be created by a small

which preserVes parabolicity.

p

If the set of uncrumpled maps of compact, then

of

S homotopic to the standard map is not

it follows immediately from the

definition that N has at least

one geometrically infinite tame end. We must show that both ends are geometrically tame. The possible phenomenon to be wary of is that the bending loci and

B;

of the two boundary components of

to a single point

A

in

jf§fls)

M:

might

8:

converge, for instance,

. (This would be conceivable if the "simplest"

9.10

homotopy of one of the two boundary components to a reference surface which

persisted in the limit first carried it to the vicinity of the other boundary

component.) To help in

understanding the picture, we will first find a

restriction for the way in which a hyperbolic manifold with

a

geometrically

tame end can be a covering space.

9.2.1. Definition. Let N be a hyperbolic manifold, P a union of horoball neighborhoods of its cusps, E' an end of N

finite-sheeted cover of end.

- P . E'

E' is (up to

is almost geometrically ting if some

a compact set) a geometrically tame

, that if E is almost geometrically

(Later we shall prove, in

tame, it is geometrically tame.)

9.2.2. Ihgg‘m . Let N Egg hnerbolic manifold , 2; N such that

N

3

5 connect set,

3

space

bounded

_cgthe imageg E i_n' N-P,

22313.

is El; almost geometrically tame end o_f. N .

in the projection to N

degree of the projection of image of

2 coverigg

has finite volume and some finite cover

Consider first the case that all points of E

SLQ: a)

N

has a_ geometrically infinite tame and E

sh: a) . Then either N fibers over S:L with fiber S ,

by 5; surface

93

N-P

and

identified with

lie in a compact subset of E . Then the local

E to

N is finite in a neighborhood of the

S . Since the local degree is constant except at the image of S ,

it is everywhere finite.

Let

G

Col N

be the set of covering transformations of

E3

over N

consisting of elements

g

bounded neighborhood of

v1 S

such that

g

. G

gEfi

E

is all of

E

except for a

is obviously a group, and it contains

E 3 up to compact sets, is an

with finite index. Thus the image of

almost geometrically tame and of N

E

r-— .~ L 1 - a l (

.

I/. (

/

El

\

1'}

‘

I

L

‘._-.—

The other case is that of

S[;,n)

is identified with a non-compact

subset

E by projection to N . Consider the set I of all uncrumpled surfaces

in E Whose images intersect the image of

an uncrumpled surface of

S[s,°)'

Any short closed geodesic on

E is homotopic to a short geodesic of E

(not a

cusp), since E contains no cusps other then the cusps of S . Therefore, by the proof of

8.8.5,

a compact closure). which fibers over

the set of images of

If I itself is not compact, than N

S1 ,

by the proof of

(since uncrumpled surfaces cut E

8.10.9.

S .

has a finite cover

If I is compact, then

into compact pieces)

componentsof the set of points identified with disjoint from

I in N is precompact (has

,

S[s a) are 9

infinitely many

compact and

9.12

A non—(ompflt‘fsur’eff .3, id6n+‘n‘Ff?J (4)!”ng (5f'

5 Surgcés {gent-gal wiflS: S These components consist of immersions of k-sheeted covering spaces of injective on

11’]- , which must be homologous to

immersions with .the same sign, homologous say to

S

ik [SJ . Pick two disjoint

- k[S]

and -2.[S].

Appropriate multiples of these cycles are homologous by a compactly supported

3-chain which maps to a 3-cycle in N theorem follows from

8.10.9.

to

a subsequence

N has finite volume. The

9.2.2.

Cont inuation £110ng 9.2. pass

- P ,hence

We may, without loss of generality,

of‘representations pi

such that the sequences of

{8:} and {Bi} converge, in 9580(8) to laminations 8+ and 8‘ . If 8+ , say, is realizable for the limit representation 9 , then any uncrumpled surface whose wrinkling locus contains 8+ is embedded

bending loci

and locally convex

- hence

it gives

a

geometrically finite end of N . The

only missing case for which we must prove geometric tameness is that

neither

9.13

8+

8

nor

is realizable.

Let

A:

CP;%(S)

(s = + , -) é

be a sequence

of geodesic laminations with finitely many leaves and with transverse measures approximating

realization of

B: 8:.

closely enough that the realization of

Also

suppose that

lim

a

in

A;

in N

Nj to

go to

Ni

is near the

The laminations

A:

in N , since their limit is

not realized. We will show that they tend toward

{AI}

in

A: s B6 in4%flo(s) .

all realized in N . They must tend toward

Imagine the contrary

A:

in the

a

e

—

direction.

- for definiteness, suppose the realizations A: in the The realization that

of

of each

direction.

a

must be near the realization in N , for high enough j . Connect

A:

by a short path

surfaces

a. o.

that for

t. near

Aid-,1:

realizing the

1.3.“6

tO ,

Si,j,t

A.

in

1.3.t

and

6830(8)

. A

family of uncrumpled

is not continuous, but has the property

Si,J,to

have points away from their

cusps which are close in N . Therefore, for every uncrumpled surface

S.

between

138:0

such that

S.

?\3

and

19.1.13

n U

S.

lejal

n

(N

—

in a homolo ical sense

( S P) is non—void.

) ,

there is some

U t

7x;

a:

9.1h

U3 be a sequence of uncrumpled

Let Y be any lamination realized in N , and surfaces realizing is a sequence

U

J

y

in

Nj

, and

S.1(j),j,t(j)

converging to a surface in N . There

of uncrumpled 5urfaces

toward

whose wrinkling loci tend

NJ.

' in

‘ ' 1ntersect1ng

8+

Without loss of generality we may pass to a geometrically convergent subsequence, with geometric limit

finite volume

-

E_ of N .

end

ckze «1 E_

.

implies that the

a:

“i

have

Each element

{ai}

a

of

N . It cannot have

for instance)

a

has a finite power

in

«1 (Ni)

bounded length in the generators of

have bounded length, so a

so by

E which is the image of

le

approximating

E_ = E (up to compact sets) . Using

si(J).J.t

5 ,

geometrically tame end

Then a sequence

property that the

°f

is covered by

(from the analysis in chapter

8.1h.2 , it has an almost the

Q . Q

is in fact in

has the

rlS

; this

le_,

and

this, we may pass to a subsequence

's which cenverge to an uncrumpled surface R

incompressible, so it is in the standard homotopy class.

in E . R

It realizes

is

+

B ,

which is absurd.

We may conclude that N has two geometrically tame ends, each of which is mapped homeomorphically toihe geometric limit

Q . (This holds whether or

not they are geometrically infinite). This implies the local degree of N -—+ Q is finite one or two

(in case the two ends are identified in Q.)

But any covering transformation a of N over Q has a power in

nlN

, which implies, as before, that a e nlN , so that N

(its square) Q .

9.2

9-15

9.3.

The ending of an end.

In the interest of

avoiding circumlocution, as well as developing our

image of a geometrically tame end, we will analyze the possibilities for

non-realizable laminations in a geometrically tame end.

We will need an estimate for the area of a cylinder in a mrperbolic 3—manifold. Given any map f :

S:L x

[O , l]

6

hyperbolic manifold, we may straighten each line obtaining a ruled cylinder with the

9.3.1.

The area

931 ruled

N , where N

—>

x

is a convex

[0 , l] to a geodesic,

same boundary.

cylinder (g above ) _ii less than the len h

E

its

Proof. The cylinder can be

Co-approximated by

a union of small quadrilaterals,

.

each subdivided into two triangles.

The area of a triangle is less than the

minimum of the lengths of its sides

(see p. 6.5).

9.16 If the two boundary components of the cylinder

C are far apart, then

most of the area is concentrated near its boundary. Let the two components of

9.3.2. Area

(C

Y1

and

Y2

denote

BC .

- Werl)

§_

e-r 1(yl) +

z(yz)

where

r

3.0

2 denotes

and

This is derived by integrating the area of a triangle in polar coordinates from any vertex: T(e) A e sinh t dt d6 1) d6 (cosh w(a)

II

J

-

mm)_ a) .

.19 m

ok The area outside a neighborhood of radius r of its far edge a

I

- r)

1 d8


n) . Clearly there is some cyclic train path through b , so T

D

b

admits a positive measure.

If r

is oriented, then each region of

on its boundary. The area of S must be

S

-

T

has an even number of cusps

hr or greater (since the

complete oriented surfaces of finite area having

x = —l

only

are the thrice

9.36 punctured sphere, for which

7Wéflo

is empty, and the punctured torus.)

If

there is a polygon with more than four sides, it can be subdivided using a

branch which preserves orientation, hence admits a cyclic train path.

The

case of a punctured polygon with more than two sides is similar.

Otherwise,

S

-

y

has at least two components. Add one branch

bl

which

reverses positively oriented trains, in one region, and another branch

b2

which reverses negatively oriented trains in another.

"tit

There is a cyclic train path through

b1

and

b2

in

r U

bl U b2

9

hence an

invariant.measure.

Now consider the case when S

-

T

has more complexly connected regions.

9.37 If a boundary component of such a region R

a train pointing away.from R

toward R

.

If R

has one or more vertices, then

can return to at least one vertex pointing

is not an annulus, hook a new branch around a non-trivial

V

homotopy class of arcs in R with ends on such a pair of vertices.

If R

is an annulus and each boundary component has at least one vertex, then

add one or two branches running across R which admit a cyclic train path.

Pm

‘T.-

If R

is not topologically a thrice punctured disk or annulus, we can

add an interior closed curve to R .

Any boundary component of R which is a geodesic c has another region

R' (which may equal R) on the other side. In this case, we can add one or more branches in R

and R' tangent to o

in Opposite directions on opposite

9.38 sides, and hooking in ways similar to those previously mentioned.

I

I

'01

.'

_L

From the existence of these extensions of the original train track, it follows that an element

75

“72727.00

of triangles and punctured monogons.

is essentially complete iff S

Furthermore, every y 6

approximated by essentially complete elements dense set has the property that the

e

Wehbeo .

-Y

We

consists

can be

In fact, an open

- train track approximation re

has

only triangles and punctured monOgons as complementary regions, so generically

avg

'I:E

has this property.

then holds for

szo

The characterization of essential completeness

as well.

Here is some useful geometric information about uncrumpled surfaces.

9.5.5.

s t‘

. (a) The sum gfthe dihedral angles along

gt; the wrinklipg locus w(S)

all

edges

tending toward §._ cusp pf E uncrumpled surface

S is 0. (The smni_s_takenipthe grogp

81

= fined 21!):

(b) The sum 9; the dihedral angles along all edges 9; w(S)

tandipg

9-39 toward fl side £5 closed geodesic y

31; T«f(S) i_s

id

, where c i_s

the ggl_e fl rotation o_f_ parallel translation around 7 . (The sign depends

on the sense of the spiralling of near

geodesics toward

y) .

39%. Consider the upper half-space model, with either the end of 2; toward which the geodesics in w(S) are spiralling at level

cusp or the c-

. Above some

(in case a) or inside some cone (in case (b) ) , S consists of

vertical planes bent along vertical lines. The proposition merely says that

the total angle of bending in some fundamental domain is

\

9.5.6.

m.

lamination

_

_i_n‘

the

sum of the parts.

\

A_.n uncrumpled surface raalizipg an essentially complete

W7? 0

_ip 1 given homotopy class

E unique._

Such

a uncrumpled

surface is totally geodesic near its cusos.

Proof:

If the surface S is not a punctured torus, then it has a unique

completion obtained by adding a single geodesic tending toward each cusp.

9.5.5 , an

uncrumpled surface cannot be bent along any of these

added

By

geodesics,

9.1+o 9.5.6.

so we obtain

If S is the punctured torus a lamination y

Complete

y

T

-p ,

then we consider first the case of

which is an essential completion of a single closed geodesic.

by adding two closed geodesics going from the vertices of the

punctured bigon to the puncture.

Ar

7’

essem‘tia H y

Complelc

COWP

1 e

9

re

\

\.

If the dihedral angles along the infinite geodesics are shown, then by

9.5.5

62 + 63 a

e2

and

e3

, as

we have

el + 62 el + 63 where

6l

is some angle.

= 0 = a =

a

(The signs are the same for the last two equations .

because any hyperbolic transformation anti—commutes with a any perpendicular line).

o

180

.

rotation around

L0

el

Thus

=

92

punctured bigon.

element

7

1.7%

7,}

—

a .11; to :7:r/ 1

u-fl’t’i

(90°“(a TL: n

9.1+l

= 0 , so an uncrumpled surface is

EWXO

totally geodesic in the

Since simple closed curves are dense in

7770?

, every

realizable in a given homotopy class has a realization by

an uncrumpled surface which is totally geodesic on a punctured bigon.

If Y

'

is essentially complete, this means its realizing surface is unique.

-

W I_f v .i.§._a_n essentially complete geodesic lamination . realized 131315 uncglupled surface U , then a_ny uncrumpled surface U' realizipg 9-5-7-

E lamination y'

near

Y ignear U .

Proof: You can see this from train track approximations. This also follows from the uniqueness of the realization of

y

uncrumpled surfaces realizing laminations converging to

a surface realizing

.

on an uncrumpled surface, since y

must converge to

9.h.h

y .

Consider now a typical path

Y t 6'

771%O .

The path

Yt

is likely to

consist mostly of essentially complete laminations, so that a family of

uncrumpled surfaces

continuous.

Ut

realizing

Yt

would he usually (with respect to t)

At a countable set of values of t ,

Yt

is likely to be

essentially incomplete, perhaps having a single complementary quadrilateral.

9.h2

Ut-

Then the left and right hand limits

and

Ut+

would probably exist, and

give uncrumpled surfaces realizing the two essential completions of

Yt

In fact, we will show that any path "generic" path in

which

.

can be perturbed slightly to give a

the only essentially incomplete laminations are ones

with precisely two distinct completions.

In order to speak of generic paths,

we need more than the topological structure of

9.5.8

Yt

Wm. 177;?

911417th Er;

777x43 .

canonical PL

(piecewise linear )

structures.

2m:

We must check that changes of the natural coordinates coming from

(pp

maximal train tracks proof for

Let

W280

y

8.59 -

8.60) are piecewise linear. We will give the

; the proof for

77132

is obtained by appropriate modifications.

be any measured geodesic lamination in

WXOS) .

Let

Tl

and

2 be maximal compactly supported train tracks carrying Y , defining and the from neighborhoods of y to convex subsets coordinate systems 'r

o1

of

En

(consisting of measures on

approximation

”wall“

name‘slul.“

0

of

Y

Tl

is carried by

and

1'1

12)

and

. A close enough train track

12 .

9.1+3 The set of measures on a go linearly to measures on

Tl

and

1'2 _.

is a maximal compact train track supporting a measure, we are done

4’2 0 ¢Zl

of coordinates

is linear near

alwazs

Otherwise, we can find a finite set of enlargements of so that every element of a neighborhood of y

oi

- the change

. (In particular, note that if y

y

is essentially complete, change of coordinates is

of the

If a

linear at

o ,

.

y )

cl ,..., a

,

is closely approximated by one

. Since every element of a neighborhood of y is carried by

2 , it follows that (if the approximations are good enough ) each ' carried ' by 11 and 1'2 . Each oi defines a convex polyhedron of the al. is 1 and

1'

1:

431

which is mapped linearly by neighborhood of

9.5.9

m.

coefficients.

y

and

¢2

, so

¢2 o (bil

must be PL

in a

.

It is immediate that change of coordinates involves only rational

In fact, with more care

integral linear structure.

77%

and

77560

can be given a peicewise

To do this, we can make use of the set

'9"

of

integer-valued measures supported on finite collections of simple closed curves

(in the case of

GI.n Z

‘7”)?0) ; .9

is analogous to the integral lattice in

consists of linear transformations of

lattice. The set

V.r

u(bi)

>

En

.

which preserve the integral

of measures supported on a given train track

subset of some linear subspace V C linear inequalities

En

mm

1

is the

which satisfies a finite number of

0 . Thus

V1

is the convex hull of a finite

number of lines, each passing through an integral point. The integral points

in U

are closed under integral linear combinations (when such a combination is

9.1.1. in U)

, so they determine an integral linear structure which is preserved

whenever

U is mapped linearly to another coordinate system.

Note in particular that the natural transformations

of‘7flgi,o

are volume-

preserving.

The structure on

(jfif

and

(?G£o

is a piecewise integral projective

structure. We will use the abbreviations

PIL

PIP

and

for piecewise

integral linear and piecewise integral projective.

9.5.10.

Definition . The rational depth of an element

of the space of rational linear functions vanishing on any natural local coordinate system. From I

9.5.8

and

Y

e'szfo

is the dimension

Y , with respect to

9.5.9 ,

it is clear

that the rational depth is independent of coordinates.

9.5.11. Egg i' iop. I_f

Y has rational depth 0 , then Y

E assentially

C omnlete .

m

For any Y

ejhޣo

which is not essentially complete we must

a rational linear function vanishing on Y . Let

T

construct

be some train track

approximation of Y which can be enlarged and still admit a positive measure.

It is clear that the set of measures on r spans a proper rational subspace in any natural coordinate system coming from a train track which carries

(Note that measures

on

T

.

consist of positive linear combinations of integral

measures, and that every lamination carried by

not carried by r.)

T

r

is approximable by one

9.145 9.5.12. figpositigg . _I;

y

e777£0_ has rational depth

1 , then either

Y i_s essentially emulate _o_r- Y has precisely two essential completions. I_n this case,

new;

A. Y Egg closed

afl

3 2n .

1r

unless

m , a_nd_a_l_J= complementgy regions have

Y

Lara i_so_nl_ygz£ fish—n with

E oriented a]:

there are two. Such

a punctured

area (S) =

_

area

21f

1m , in which case

5 region i_s fie;g fladrilateral g

bigon .

Or 9; B . Y

has precisely one closed leaf

YO

has area

l.

S i_s a punctured torus

or 2.

Yo (a)

YO

. Each region touching

21r . Either

touches two regions , each

or (b)

a on

pointed crown £5devils cap. or (c)

9.h6 Suppose Y has rational depth 1 and is not essentially complete.

Let

'r

set

Tl

be a close train track approximation of

,..., Tk

approximate and let

V0,

of essentially complete enlargements of

every

Y'

in a neighborhood of Y . Let

be its coordinate system. The set of

measures carried by a given proper subtrack of a

space of

Va

. Since

set of measures

(If

Y . There is some finite

Vt.

Ti

which closely

a carry all the

carried on any

Ti

is a proper rational

positive on all branches of

Ti

VT

Y would come from a. measure

) . Since this works for any degree of

approximation of nearby laminations, Y

9.5.h

sub—

V-r , the

must consist of one side of

VT. intersected both sides, by convexity

ri's

corresponding to

is in a unique proper rational subspace,

Y

A review of the proof of

Y

t

has precisely two essential completions.

gives the list of possibilities for

Y€7W°£O

with precisely two essential completions. The ambiguity in the essential completions comes from the

manner of

and the direction of spiralling

dividing a quadrilateral or other region,

around a geodesic.

9.h7 Remark . There are good examples of Y but are essentially

complete. The

6777090

which have large rational depth

construction will occur naturally in

another context.

We return to the construction of continuous families of surfaces in a hyperbolic

3—manifold. To each essentially incomplete Y e

jfifi£o

- parameter

Us

depth 1 , we associate a l

family of surfaces

of rational

, with

U0

U1 being the two uncrumpled surfaces realizing Y . US is constant where U0 and U1 agree, including the union of all triangles and punctured and

monogons in the complement of

S

—

Y

Y . The two images of any quadrilateral in

form an ideal tetrahedron. Draw the common perpendicular p to the two

triangulate the quadrilateral with h triangles by l ’ adding a vertex in the middle, and let this vertex run linearly along p',

edges not in U 17 U

0

from

U0

to Ul . This extends to a homotopy of S

The two images of any punctured bigon in

the generating curve parabolic.

S

Y

straight on the triangles

form a solid torus, with

The union of the two essential completions in

this punctured bigon gives a triangulation except in a neighborhood of the

9.h8 puncture, with two new vertices at intersection points of added leaves.

“!vA

‘

“

Draw the common perpendiculars to edges of the realizations corresponding to these intersection points, and homotope

U0

to

Ul

by moving the added

vertices linearly along the common perpendiculars.

When

Y has a closed leaf

Yo

have added leaves spiralling around homotoped to

Ul

, the two essential completions of Y

Y0 in opposite directions. Uo

through surfaces with added vertices on

Y0

.

can be

9.19 Note that all the surfaces

Us

point on

Us

constructed above have the property that any

Us

is in the convex hull of a small circle about it on

i -l : curvature -l

particular, it has curvature

. In

everywhere except

singular vertices, where negative curvature is concentrated.

m.

9.5.13.

bnerbolic 3—manifold

Given any complete

s[s a) , there _i_s_g proper g s 39 .. i_nE.

tame and E cut off _‘qya merbolic surface

m. in

WAD

the

Yi

segment

s[8,a)x[o,o)~

F:

homotopy

VT

Let

0(8)

9

be a natural coordinate system for a neighborhood of

, and choose a sequence

slightly. so that the path

te [i , i+l]

0 or 1 . Let

Yt

N

N with geometrically

Ut

a(E) . Perturb

limiting on

[O _

S' has only one sheet

J

- it is

a homeomorphism.

Let Q be the geometric limit of any subsequence of the

Ni

. N is a

Q . Every boundary component of the convex hull M

covering space of

Mi

is the geometric limit of boundary components of the

of N

; consequently, M

covers the convex hull of Q . This covering can have only finitely many sheets, since M

-P

is made of a compact part together with geometrically

infinite tame ends. Any element

[k 1 l]

a, E

le

has some finite

power ck E

1T1N

. In any torsion-free subgroup of PSL (2 , E) , an element has at

most one

kth

of elements

(by consideration of axes). If we write a as the limit

root

pi(gi)

constant so a

, gig

r1 N0

,

by this remark,

is actually in the algebraic limit

converges strongly to

p

gi must be eventually

wlN

. Q = N , and pi

.

Ni

A cusp-preserving homeomorphism from N to some

, hence to

No

,

can be constructed by using an approximate isometry of N' with a sub—manifold of

to

Ni - Pi

N1

, for

high enough

i . The image of N'

is homotopy equivalent

, so the fundamental group of each boundary component of N' must

map surjectively, as well as injectively, to the fundamental group of the neighboring component of into

Ni

(Ni , Pi)

- N' .

This implies that the map of N'

extends to a homeomorphism from N

There is a special case remaining.

constructed in N

-P

to

Ni

.

If any pair of the surfaces

is homot0pic, perform all such homotopies.

Si

Unless

N

-P

9-58

8N0 - Po .

is homotopy equivalent to a product, the argument continues as before is no reason the cover of

When N then by 8

-P

Si

must be a connected component of

standard

argument

NO - P0 must be homeomorphic to Si 9.2 .

the case essentially dealt with in

ends of N

-P

Si

is homotopy equivalent to the oriented surface

laminations of the two ends of

Ni - Pi

in it,

I . This is

The difficulty is to control both

9.2

7 but the argument of

X

there

shows that the ending or bending

cannot converge to the same

x1

lamination, otherwise the limit of some intermediate surface would realize

.

9.6.la

59.7.

nggizatiggs _of geodesig Emily :9; surface gpoups with extra cusps,

with_a.

WW— on Minates .

In order to analyze geometric convergence, and algebraic convergence in more general cases, we need to clarify our understanding of realizations of geodesic laminations for a discrete faithful representation

group

«1(8)

Let N =

when certain non-peripheral elements of

Np«ls

be the quotient

"top"

N+

y_ be the (possibly empty) cusp loci and

51-

,..., Sk-

the components of

-P ,

where

are parabolic,

§8.ll , we may embed S in N ,

and the

for

S

N+

- 7+

hyperbolic structures with finite area). Let denote the ends of N

"1(8)

of a surface

3-manifold. Equip S with a complete

hyperbolic structure with finite area. As in cutting it in two pieces, the

9

"bottom" N_ . Let

and

N_ , and

and S

Elr

- Y-

,..., EJ+

Y+

denote by

and

81+ ,..., Sj+

(endowed with complete and

El—

,..., Ek-

P is the union of horoball neighborhoods

9-59 of all cusps.

A compactly supported lamination on

e(Ei+)

S . In particular,

Si+

or

Ei+

S for

.

W . g lamination yeiiow) contains g cammnent g Y+ . a component E c(Ei_) . 9.7.1.

If y

defines a lamination on

may be thought of as a lamination on

each geometrically infinite tame end

Proof.

Si—

i_s realizable i_g N Y_.

.

iff y

and-a a(Ei+) gr;

contains any unrealizable lamination, it is unrealizable,

so the necessity of the condition is immediate.

Let

y'

c’7NE?o(s)

be any unrealizable compactly supported measured

lamination. If y is not connected, at least one of its components is unrealizable, so we need only consider the case that

y

has zero intersection number with any components of

7+

is connected. If y or

y_ , we may cut

S along this component, obtaining a simpler surface S' . Unless y component of

7+

covering space of

of

or

y_

in question,

N corresponding to

S' are parabolic, so we

is the

8' supports y , so we pass to the

NlS'

. The new boundary components

have made an inductive reduction of this case.

We may now suppose that y has positive intersection number with each component of

7+

and

y_ . Let

{8i}

be a sequence of measures, supported

on simple closed curves non-parabolic in' N which converges to Y . Let

{Ui}

be a sequence of uncrumpled surfaces realizing the

Si

. If

Ui

penetrates

9.60 far into a component of P

corresponding to an element

a

in

7+

or y_ ,

then it has a large ball mapped into P ; by area considerations, this ball

on

Ui

of c

must have a short closed loop, which can only be in the homotopy class

.

Then the ratio

£S(Bi)/ i(Bi,

a.)

is large. Therefore (since i(y , a)

Ui

iUi(Bi)/i(Bi,c)

;

is positive and

18(7)

is finite) the

, away from their cusps, remain in a bounded neighborhood of N

N . If

Y+

so that any

-P

in

(say) is non-empty, one can now find a compact subset K of N

Ui

By the proof of

intersecting

8.8.5 ,

P

intersect K

if infinitely many

Ui

intersected K , there would

be a convergent subsequence, contradicting the non-realizability of y . The only remaining possibility is that we have reached, by induction, the case that

either

N+

or N

has no extra cusps, and

y

is an ending lamination.

Y€i& (S)

A general lamination

is obtained from a possibly empty

lamination which admits a compactly supported measure by the addition of

(Let 5 C Y be the maximal lamination

finitely many non-compact leaves.

supporting a positive transverse measure.

5 or go to

each end must come close to

enlarge

If w

2. is any leaf in Y

in

is realizable iff

5 is .

The picture of unrealizable laminations in

A+

S , otherwise one could

6 . By area considerations, such leaves are finite in number.) From

58.10 , Y

Let

-6,

@f o(S)

is the following.

consist of all projective classes of transverse measures (allowing

x+ = Y+ U U 1 a(E.1+)

degenerate non-trivial cases) on a coordinate system

VT

coming from any train track

.

1:

A+ is convex.in carrying

x+

.

To see a larger, complete, picture, we must find a larger natural coordinate system. This requires a little stretching of our train tracks and imaginations.

In fact, it is possible to find coordinate systems which are quite large. For any

Y€

(1pr ,

let

AY C. 6’30

denote the set of projective classes of

measures on Y

9.7.2.

Mam .

of train tracks

—- natural of

Ti ,

Let

where

coordinate systems

b_e essentially complete . There i_s_a_

Y

i_s_ carried by

Ti

5Y

= Ui v11

‘L'i+1

sequence

, such that the union

— 6765 0 -

contains all g

A 'Y .

The proof will be given presently.

Since

I. 1

is carried by

1+1 , the inclusion V Ti C V Ti+l

1'.

is a projective

9.62 map

(In

Wflo ,

the inclusion is linear) . Thus

SY

comes naturally

equipped with a projective structure. We have not made this analysis, but

AY

=

the typical case is that Y

We think of

SY

as a stereographic

coordinate system, based on projection from Y . (You may imagine as a convex polyhedron in

IRn

, so that

WEE/O

changes of stereographic coordinates

are piecewise projective, although this finite-dimensional picture cannot be strictly correct, since there is no fixed subdivision .sufficient to make all

coordinate changes.)

’Y

57 9.7.3. Corolla. will be computed in

Proof.

65f. o(S) )

t_oa Sphere. (whose dimension

_i_s_

.

(9.7.29 9.7.3) Let Y

lamination. Let

E‘ 9.7.h) The only projective structure on

n > 1 , is the standard one, since

Sn

is simply connected.

S11

, when

The binary

relation of antipodality is natural in this structure. What would be the

antipodal lamination for a simple closed curve

at o ,

Remark

the Dehn twist around

When

? It is easy to construct

but moving any other given lamination.

a diffeomorphism fixing a i (Y , a)

a

63» O(S)

quadruply punctured sphere)

is

, the

a. will do.)

l—dimensional

PIP

(If

9.7.h.

(S = punctured torus or

structure does come from a projective

2P1 . The natural transformations of 073‘ o(S) are necessarily integral - in PSL2 ( Z ) . structure, equivalent to

Proof region

Ri

IR

g 9.7.2. Don't of

S

blink.

Let Y be essentially complete. For each

- Y , consider a smaller

with finite points, rotated so its points

very slightly through the sides of

Bi

,

region

ri

of the same shape but

alternate with cusps of ending on a leaf of

Y .

R1

and pierce

9.6h

fi—L

By

9.5.h , 9.5.2

so the regions

S

- Y - Uiri

ri

and

ri

9.3.9 ,

both ends of each leaf of

separate leaves of Y into arcs. Each region of

muSt be a rectangle with two edges on

covers the "interesting" part of

Bi

Collapse all rectangles, identifying the

a surface

I

3r;

and two on Y , since

. (Or, prove this by area, x) .

ri

edges with each other, and obtain

S' homotopy-equivalent to S , made of

to a train track

Y are dense in Y ,

Uiri

. (Equivalently, one may think of S

, where

- Uiri

ari

projects

as made of

very wide corridors, with the horizontal direction given approximately by

Y) .

9.65 If we take shrinking sequences of obtain a sequence of train tracks

T.)

Tk

carries

when

j > k

13

. Let

regions

ri,j

in this manner, we

which obviously have the property that

Y' e

6:. o(S) - AY

be any lamination

not topologically equivalent to Y . From the density in Y of

Y , it follows that whenever leaves of Y

and

Y'

of ends of leaves

cross, they cross

at an angle. There is a lower bound to this angle. It also follows that

Y

LIY'

cuts S into pieces which are compact exc

S .

for cusps of

”Y

9L:

7' When

R1

is an asymptotic triangle, for instance, it contains exactly one

‘region of S

- Y - Y'

which is a hexagon, and all other regions of S

are rectangles. For sufficiently high changing the leaves of

follows

’Y

F

Y'

j , the

r13 can be isotoped, without

Y which they touch, into the complement of

projects nicely to

- Y - Y'

Y'

. It1

rJ

’Y 7' 9.7.2. I

9.66 Stereographic coordinates give a method for computing and understanding

intersection number.

measure

Y

The transverse measure for Y projects to a "tangential"

vY on each of the train tracks

Ti

- transverse length of the sides of the

: i.e., u (b) is the Y

rectangle projecting to

b .

b

It is clear that for any a a

777£0

which is determined by a. measure

u“

011':

i

9.7-3

i(a

.Y)

Thus, in the coordinate system

I):

ua(b) -vY(b)

VT i

in

Wg-o

, intersection with Y is a

linear function.

To make this observation more useful, we can reverse the process of finding a family of

"transverse" train

tracks

Ti

depending on a lamination Y .

9.6? Suppose we are given an essentially complete train track

, and a non-negative

T

function (or "tangential" measure) v on the branches of b , subject only to the triangle inequalities

a +b

Whenever

_

in

S .

-c

a + c

a , b and c

-r .

regions

3_ O

-b

O

b + c

-a

3_ 0

are the total v-lengths of the sides of any triangle it

We shall construct a "train track" 1 dual to r , where we permit ' * 1 to be bigons as well as ordinary types of admissible regions of S

let us call

a

1

—

track.

a

Q t l LLLT\

fitsjturecj

SP‘he‘re Ta:

*-

1

is constructed by shrinking each region

obtain a region points are

branches

R: i

—)

on.

on.

"transverse"

. If we drop the restriction that the measure

9'72 on

Ti

on

1:j

on

1'1

is non-negative, still it often pushes forward to a positive measure

.

The image of

SY

is the set of such arbitrary

which eventually become positive when pushed far enough forward.

For

Y' e

AY

9.7.6. W .

, let

vy,

-

1: >

be a "tangential" measure on

11 pp

0 .

(Note that the functions

vY,

In particular, note that if coordinates for

sY

The image Q

necessarily positive , measures

vY

"transverse" measures

7”)? 0

-

Tl

“Y"

AY

= Y

-

k-simplex, then the image of

int

(Ak)

x

u are distinct

AY

for

,

Y' a! Y"

$7

for

We}; 0

fid'fo

the image is

IRn

Ime—

Emu-36 SA“-

The condition that

intersection number

If

(This image is defined only up to projective equivalence,

L

Proof.

.)

is of the form.

until a normalization is made).

711A”?

not

, the image of stereographic

is a half-space, or for

is a

Y'g

Y'

defining

939.11"transverse'' ,

such that forall

u and

AY

Ian-k .

i_sthe set

Tl _

VY'

i(Y' , Y") for

-u Y

>

e

0 is clearly necessary:

AY

Y" 6

SY

is bilinear

9.73 and

given by the formula

i(Y'

vY

.Y") u

Consider any transverse measure

Ti

nonrpositive when pushed forward to

such that the push-forward of

for high

u

HY" on

.

11 Let

u

such that

bi

be a branch of

is non-positive on

bi

i , comes from a very long and thin rectangle

pi .

average transverse counting measures of one of the sides of

pi in a natural way to

of the

pi . To make

13*

for 1

j

5.1

.

carries

in a narrow corridor around 1 , so that branches of a a is obtained by squeezing do not pass through switches of o . Now

To see this, embed a

bi

'There is a

limit

(In general, whenever an essentially complete train track l* a train track carries T 0 , then a

I

Ti

. This branch

standard construction for a transverse measure coming from a

this more concrete, one can map

is always

a

all intersections of branches of

*

T

.

with a Slnglebranch of

.

o to a Single

point, and then eliminating any bigons contained in a single region of

S

- o .)

9.7h

1:1

On

, pi

pi

times

is a finite but very long path.

The average number of

it

Tl

tranverses a branch of

gives a function

vi

which almost

satisfies the switch condition, but not quite. Passing to a limit point of

{vi}

one obtains a

-

v

single branch

CY

bi

at

Ti

of

Y'

ZY

. The

#3? 0(5)

(bi)

u

, and

consist of laminations

arbitrary element of

Y

Y'

i(Y , Y') = 0

such that

not intersecting

Y

a

Y

is disjoint from the support of

i-e-a

. An

CY , together with some measure

is an element of

c untaining

Y

is a simple closed curve and

of

Y'

consist of

ZY

withgygfngg canonical

i_s convex . (In

Suffices to give the proof in

X

777360

«WHQO . First

= Vr ,

or $660) . consider the case

for some train track

Y . Pass to the cylindrical covering space C of

group generated by track

, for all i

.

Pm;- It carrying

1;?-

same symbols will be used to denote the images of these sets in

coordinatesystems X

Y

, whose lamination topo-

0 .




i(I) and

_>

i(I)

w(I)

for all subintervals I ‘

_jZn.

Tenibme‘mlei'r arTcron—le‘H' _‘

Q >>O

‘

L-hX

l-n

E‘s-(ea dum

r

\F

l

I

L


0 suchthat

sure snace

9".

T1(M)

t>T,a_.n_cl

is recurrent.

x;\.16¢t(A) (A flow

¢t

and a

.)

on a mea-

(X,u) is recurrent when for every measurable set A.C:X. pf

measure and

..(A ߢt(A)) >

every

T

>

0 there is a t

>

T such that

o .>

(d) The geodesic flow on

T1(M)

ergodic

Note that in the case M has finite volume, recurrence of the geodesic flow is immediate (from the Poincare recurrence lemma). The ergodicity of the geodesic flow in this case was proved by Eberhard Hopf, in [Hopf, The idea of

] .

(c) ———+ (d) goes back to Hopf, and has been developed more

generally in the theory of Anesov flows {Anosov,

] .

9.9-2.

9.9.2.

If the geodesic flow is not

Corollgz

ergodic.

there is a

non-constant bounded suoerharmonic function on M . Proof of 9.9.2. _ Consider the Green's function g(x) = ,

for hyperbolic space.

By (a), the series which projects to (where arctan

a

(This is a harmonic function

2

g o y

1-

.

813

d(x,xo)

blows up at

nt dt

h

no .)

converges to a function, invariant by y ,

afar En's function

= n/2)

which

I.

G for M . The function f = arctan G

is a bounded superharmcnic function, since arctan is

convex. Remark

The

convergence of the

series of (a) is actually equivalent to

the existence of a Green's function on M ,

and also equivalent to the existence of

a bounded superharmonic function. See [Ahlfors and Sario,

case n = 2 , and

IE

] for the general case.

909'3. Corollfl.

E r is a geometricallz tame Kleinian geodesic flow on T1(sn/r) is ergodic iff LT = 82 . Proof

9.9.3.

] for the

From 9.9.2 and

8.12.3.

gone, the

9.9.3.

B

Sullivan's proof of 9.9.1 makes use of the theory of Brownian motion

on

MP

. This approach is conceptually simple, but takes a certain amount

of technical background (or faith). Our proof will be phrased directly in terms of geodesics, but a basic underlying idea is that a geodesic behaves like a random path:

its future is "nearly" independent of its past.

r

9.9—2a.

.l I«

fil

r

J! ‘. 5"“: 555644”

,.

Pa ’C 5

Geodesics with nearly identical pasts can have very different futures.

r;,LL-lUv“ - c24.

9.9-3. (d) -* (c). This is a general fact. If a flow

measure,

Li ¢t(B)

section with A

is not recurrent,

of positive measure such that only for t in.scme bounded

there is some set A

interval is u(At1

¢t

¢t(A))

>

0 . Then for any subset B C A of small enough

is an invariant subset which is proper, since its inter-

is proper.

(c) —* (b). Immediate.

(b) —r (a). Let B be any ball in

where

r = “1M . For the

an , and consider its orbit

r3

series of (a) to diverge means precisely that the

x0 3 En

total apparent area of PB as seen from a point

, (measured with

multiplicity) is infinite.

In general, the underlying space of a flow surable parts, X

= D U R , where ¢t

is decomposed into two

mea-

is dissipative on D (the union of all

subsets of X which eventually do not return) and recurrent on R . The reader may check this elementary fact. If the recurrent part of the geodesic flow is non-empty, there is some ball B in

M11

sure of the tangent vectors to points of B

give rise to geodesics that intersect B

infinitely often.

such that a set of positive mea-

This clearly implies that the series of

(a) diverges.

The idea of the reverse implication (a) -> (b) is this: if the geodesic flow is dissipative.there are points

x0

such that a positive pro-I

portion of the visual sphere is not covered infinitely often by images of

some ball. Then for gagh "group" of geodesics that return to B, a definite proportion must eventually escape rB , because future and past are nearly independent. The series of (a) can be regrouped as a geometric progression,

so it converges. We now make this more precise.

Recall that the term "visual sphere" at rays" emanating from

x0.

x0

is a synonym to the

"set of

It has a metric and a measure obtained from its iden-

tification with the unit sphere in the tangent space at

x0 ,

9.9-4. Let

x0 a Mr1

be any point and B C

portion of the rays emanating from

1:0

M‘n

any ball.

If a positive pro-

pass infinitely often through B ,

then for a slightly larger ball B' , a definite proportion of the rays emanating from 21 point

x e

M11

spend an infinite amount of time in B' ,

since the rays through 1 are parallel to rays through

a subset of

T1(B')

x0

. Consequently,

of positive measure consists of vectors whose geodesics

spend an infinite total time in

T1(B')

; by the Poincare

recurrence lemma,

the set of such vectors is a recurrent set for the geodesic flow. (b)

holds so

(a) ——r(b)

is valid in this case.

To prove (a) —* (b),

it remains to consider the case that almost every ray from

1:0

eventually

escapes B ; we will prove that (a) fails, i,_e,—.,-_t'h.e. series of (a) converges. Replace B by a slightly smaller ball. Now almost every ray from al—

most every point x e M eventually escapes the ball. Equivalently, we have a ball B C through at

an

such that for every point x

intersects I'B , or even

P(NE(B))

5

En

, almost no geodesic

, more than a finite number

of times.

Let x0 be the center of B and let a be the infinum, for y 2 En , of the diameter of the set of rays from xO which are parallel to rays from y which intersect B . This infinum is positive, and very rapidly approached

as y moves away from

1:0 .

9.9 5 Let R be large enough so that for every ball of diameter greater than a

x0

in the visual sphere at

intersect

I'NE(B)

, at most (say) half of the rays in this ball

at a distance greater than R from

x0

. B should also

be reasonably large in absolute terms andin comparison to the diameter of B.

Let x0 be the center of B . Choose a subset I'l c I‘ of elements such that! (i) For every 7 e I‘ there is a ‘1'. e I" with d(y'xo,yx0)

0 .

'31

and

Let us consider what I must look like near aa

B1 x 32 .

If

y

is a "large" element of

1'

such that 7x is near x , then. the preimage by Y of a product of small

e-ball

around Yxl

and

of one of the factors," in one direction mately

:2 .)

7x2 (x1 x

is

one of two types: it is a thin neighborhood

32) or

(231 x x2)

or the other along an

Since

.

(Y must be a translation

axis from approximately

I'. is recurrent, almost every point x a

1.1

to approxi-

El x 32

-s

the

9.9-8. preimage by elements 7 of both types, of an infinite rather of points where

I‘ has density 0 or 1 . Define

x1

f(xl) =

[32 xICnyxz) dx2 _, who...

is the characteristic function of I, for

:1 a B1 (using a.

on B ).' By the above, for almost every a, there are arbismall. intervals aromd xi such that the average of f _nthat in-

probability measure trarfly

terval is either 0 or .1 . Therefore f is a characteristic function, so

I 0 Bl

x

B2

is of the tom

3

x

32

(up to a set of measure zero) for

someset

sc:1al . Similarly,'i'.- isofthefon

(In; to a set of measure zero).

]l A[

B. XB,_so

31x32 9.9.1 U

I is either ¢>

Tx (H3) .

coming from orthogonal projection

to take care of covariant tensor

fields, like vector fields, and contravariant tensor fields, like differential forms and quadratic forms, aswell as tensor fields of mixed type. The visual average of any tensor field T

on

H3 .

on

Si

is thus a tensor field

In general, avT needs to be modified

by a

avT, of the same type,

constant to give it the

right boundary behaviour.

We need some formulas, in order to make computations in the upper half-space

Let x be a point in upper half-space, at Euclidean height h above

model.

the bounding plane hits

a

a.

A geodesic through x at angle 9 from the vertical

at a distance r = h cotangent (9 / 2) from the foot

perpendicular from x to

c.

.3

20

of the

Thus, dr = sphere at

- mlo‘ csc2 56 d9

x to

SE

u

h

is Lebesgue measure on

Any tensor T at the point 9

S;

= 0'.

by the maps

9(-

1

and

2

r -2 (h+—h) do (1'.

Vx

and

is visual measure at

x

x pushes out to a tensor field

Tm

on

X is a vector, then

Xan

is a

p* . When

holomorphic vector field, with derivative

TX

Since the map from the visual

is conformal, it follows that

de where

- % (h + 1'?) d9 .

=

iHX“

at its zeros. To see this, let

be the vector field representing the infinitesimalisometry of translation

in the direction X . The claim is that geometrically when

X

X”

=

is at the center in the

TX I S0° .

Poincare

This may be seen

disk model.

K a. flxis circ‘e \

PP.

'

(01

‘->.

\,

u

Alternatively, if X

§

.n

_‘i

aT-ra ;:1

ra‘

Q

‘5

is a vertical unit vector in upper half space, then we can

compute that

a . X..=-Sm9m= where

20

h

sine

§.—2— Sin 9/2

orB

B

B

In

is the foot of the perpendicular from x

(Z'zo)ez':

to C: . This clearly agrees

with the corresponding infinitesimal isometry.

(As a "physical" vector field,

is the same as the unit horizontal vector

,

ield

this notation is that the differential operators

on holomorphic functions:

g , a: § 3% on

and

§z

. The reason for have the same action

they are directional derivatives in the appropriate

Even though the complex notation may at first seem obscure it is useful

direction.

because it makes it meaningful to multiply vectors by complex numbers.)

=

where

Tx(H3) ,

g _is the standard inner product on

When

Y1 Y2 '

then

2 -2

r_

1

.

is the izmer product of two vectors on

a

m

.

Let us now compute av ( F ). By symmetry considerations, it is clear that .

av (

g2 )

is a' horizontal vector field, parallel to

8;

vector of unit hyperbolic length, parallel to

g

. Let

at a point

e be the

x in upper half-

space. Then

em: --2£h-(Z-Zo-h)(Z-Zo+h)

382'

e

e /.\

L

Wehave

3 5V3;

$1;er (5;)av $2

soavsa; e‘fifge,

=EfRe(-:(Z-ZO)

'3)de

CL

1

1

2 'h)16 2

2 -1+

r_ (h+h)

d“

(1

Clearly, by symmetry, the term involving Re (z

—

2 .

zo)

.

integrates to zero, so we ha'.

%ff

av(a)

8h

O

0

r2-3m

2h2

(h+—)l h o

i

bul .)

(

we

h

§z

Note that the hyperbolic norm of av

goes to

99

as h

O , while the

—-)

Euclidean norm is the constant 2

3

We now introduce the fudge factor by defining the extension of a vector field

X on

82

to be

an

ex (X) and

arm 11.1.1

E

Proposition.

X

E

When

av(X) in

x

on

H3

s:

E continuous

X

ho lomornhic , then ex(X)

Proof.

=

%

or Lipschitz ,

, then so

ex(X) .

is an infinitesimsl isometry.

X is an infinitesimaltranslation of E , then ex(X) is the

1

same infinitesimal translation of upper half-space. Thus every "parabolic" vector fiel:

,-;

i. a zero of order 2) on (With

--..._.__._---

J

9 S;

_

extends to the correct infinitesimal isometry.

A general holomorphic vector field on

Si

_

_ _

is of the form

9

_.

(az' + bz + o)

$-

on

(I:

"Since such a vector field can be expressed as a linear combination of the parabolic '

.

vector fields

o—za

,

2

2

a F2

and

(z

- 1)2 Fz'a

.

.

it follows that every holomorphic

vector field extends to the correct infinitesimal isometry. Suppose X

H3

is continuous, and consider any sequence

converging to a point at

on

. Bring

xi

[

xi

} of points in

back to the origin 0 by the translati;

.

1.1.6

Ti

along the line

a:

. If

neighborhood of the endpoint yi

all the sphere-

11* X

point to yi

, so

that for any

8

”11,3 - Pill s ).i

where

.

s:

is close to

Ti

,

from

of the geodesic

spreads a small

O to

xi

over almost

is large on most of the sphere, except near the antipodal

it is close to a parabolic vector field

, and

s

xi

Pi ,

in the sense

1,

sufficiently high

T-LX "L

- xi ,

is the norm of the derivative of

parabolic vector field agreeing with

Ti*X

Ti

Pi

at yi . Here

is the

at yi , and O at the antipodal

point of yi . It follows that

exX so X

BB3 .

is continuous along

you see

the. evidence).

Suppose now that X

(xi)

-

X(yi)

-—-)

0 ,

Continuity in the interior is self-evident

is a vector field on

SE

C

E3

(if

which has a global

Lipschitz constant

k =

Then the translates

11* X

vmr' GS

“H

X

- Xy H l

y

v

-y'

H

satisfy

H where

sup 2

Ti, x

Pi ll 5

B

B is some constant independent of i . This may be seen

by

considering

stereographic projection from the antipodal point of yi . The part of the image

of X

11*

- 1:73; P. 1

i

in the unit disk is Lipschitz and vanishes at the origin.

is applied, the resulting vector field on

When

C: satisfies a linear growth

condition (with a uniform growth constant). This shows that, on

JI [

52

H 71* X - Pi”

,

is uniformly bounded in all but a neighborhood of the

,

antipodal point of Y

ex r,* l

at the

-

P.(x.)” l l -< B3 u.IL , where u.1 is the norm origin in B , or 1 / Xi up to a bounded

“i

Since

it follows that

is on the order of the

1'

Si .

be a translation as before taking x to O

vector field approximating

1*X .

xi

B3 ,

1*

B3 .

that

x . Since the vector fields

'

7;]- P

Si

consider x e B3

,

obtained in this

ex(1'*X

- P)

By comparison with the

- 1,11 P)

at an arbitrary

are uniformly Lipschitz, it follows

Si

(l

of a uniformly Lipschitz vector field

is not necessarily uniformly Lipschitz

This is explained by the

example.

on

,

X is globally Lipschitz.

Note that the stereOgraphic image in on

l

can be taken to be a similarity, we obtain

a uniform bound on the local Lipschitz constant for ex(X point

171

from yi

way are uniformly bounded, so it is clear that the vector fields

upper half-space model, where

-

, and P a parabolic

1*X - P

The vector fields

have a uniform Lipschitz constnat at the origin in

X(xi)

factor.

(Euclidean) distance of

ex X is Lipschitz along

ex

of the derivative of

To see that ex X has a global Lipschitz constant in and let

H

Then

where boundedness is obvious.

and on

field on

B3

large

0'. near the point at

an

—

consider

H3 ,

,

for

deviation of the covariant derivatives

. Similarly, a uniformly Lipschitz vector

is not generally uniformly Lipschitz on

the curvature of

22 3%

H3 .

In fact, because of

a uniformly Lipschitz vector field on

E3

must be bounded;

3

such vector fields correspond precisely to those Lipschitz vector fields on B which vanish on

633 .

’PQV'Q\‘e"

J

51

HQ.

11.8

A hyperbolic parallel vector fiel; along a curve near S°° appears to turn rapidly.

°°

eodcsk The significance of the Lipschitz condition stems from the elementary fact that Lipschitz vector fields are uniquely integrable. Thus, any isotopy

of the boundary of a Kleinian manifold

ht

derivative

' /

to an isotopy ex ‘

proog that

.9

,

OF- = (B3 - LP) / P

is Lipschitz as a vector field on

¢t

on

of .

IX

whose time

BOP

extends canonically

One may see this most simply by observing that the

ex X is Lipschitz works locally.

A k-guasi- isometric vector field is a vector field whose flow,

distances

njmff must

mt

¢t

distorts

at a rate of at most k . In other words, for all x , y and t satisfy

-kt e

d(x,v)

S d(vtx,¢tv) 5

en d(x,y)

A' k-Lipschitz vector field on a Riemannian manifold is k-quasi-isometric. In fact, a Lipschitz vector field X on

isometric as a vector field on of the origin in

B3 .

H3

To see this

B3

which is tangent to

int

B3 .

for an

BB3

is quasi-

This is clear in a neighborhood

arbitrary point

x , approximate X

near x by a parabolic vector field, as in the proof of 11.1.1, and translate

x to the origin.

In particular, if then ex

Qt

¢t

is an isotopy

of

BOP

with Lipschitz time derivative,

has a quasi-isometric time derivative, and

$1

is a quasi-isometry.

Our next step is to study the derivatives of ex X , so we can understand

how a more general isotopy such as

the definition of

T is an If X

distorts the hyperbolic metric.

From

ex X , it is clear that ex is natural, or in other words,

isometry of

T, (ex (10)

(53* X)

ex when

(pt

ex

H3

(extended to

SE

where

appropriate.)

is differentiable, we can take the derivative at

ex[Y,X]

[Y,exX]

for any infinitesimal isometry Y . If Y point on the axis of

vXYx

Y , then

VZW

covariant derivative, so

in the direction of the vector

[Y

T = id., yielding

is a pure translation and

X is any

= O . (Here, V is the hyperbolic

is the directional derivative of a vector field W

Z .) Using the formtfln.

va - VXY

, x]

we obtain

11.1.2. Proposition. The directional derivative o_f ex X in the direction fiapoint

er3,§ V

Yx

where

Yx ,

exX = ex[Y,X]

Y i_s fix grim—e_maltim—lmn with axis throgh x and value

Yx

at x.

The covariant derivative V

tangent space

Tx(H3)

Xx

,

which is a linear transformation of the

to itself, can be expressed as the sum of its symmetric

and antisymmetric parts ,

11.10

VSX

VX where

and

Va'x,

stx

r - El (VYx r

V; x

Y'

The anti-symmetric part

VaX

=

+

E1 (V& x r'

VY.

x

V&,

X

I

r)

Y)

describes the infinitesimal rotational affect of

the flow generated by X . It can be described by a vector field curl X

pointing

along the axis of the infinitesimal rotation, satisfying the equation

vgx where X

-1 — 2curlXXY

is the cross-product.

If

eo , e1 , e2

forms a positively oriented

orthonormal frame at X , the formula is

Z

curl X

(V i X

vei+l X - ei) ei+2

e.

i e Z /3

Consider now the contribution to area on

SE ,

centered at

horosphere about y fixing y

is the

ex X from the part of X

y . This part of

on an infinitesimal

ex X has constant length on each

(since the first derivative of a parabolic transformation

identity) , and it scales as

e-3t

,

t

where

is a. parameter

(Linear

measuring distance between horospheres and increasing away from y .

measurements

scale as

e-t

. Hence, there is a factor of

scaling of the apparent area from a point in

the scaling of the lengths of vectors.)

t , x1

, x2 , so

that

3x1

1

describing the

H3 , and a factor of e-t

representing

Choose positively oriented coordinates

use = at2 + .22t (dxi + dx:) ,

contribution to ex X is in the

e-2Jc

direction.

Let

and this infinitesimal

eo , el

and

e2

be unit

vectors in the three coordinate directions. The horospheres t = constant are parallel surfaces, of constant

7‘1

'7

233) ,

normal curvature 1 (like the unit sphere in V7

= V’

Ve1 eO

=

e0 e0

V

and

e9

e0

so you can see that

= V’

e0 e1

0

e0 e2

+e1 ’ Vt e1

ea ’Ve9 e2

-e0 Vt1 e2 =

-e0 ’Ve2 e1

0

0

(This information is also easy to compute by using the cartan structure equations.) The infinitesimal contribution to ex X is proportional to

Z =

e'3t el

curl Z =

_—

, so

Cve0 Z

'

el

,

-

V;

l

Z

e0 e2

-2 e ‘3t e2 .

(The curl is in the opposite sense from the curving of the flow lines because the effect of the flow speeding up on inner horospheres is

This is proportional

iX to

the contribution of

stronger.)

ex iX from the same

infinitesimal region, so we have

11.1.3 Propgsition. Curl

(ex X)

2 ex (iX) ,

Curl2

(ex X)

-h ex X

and consequently _l

.J

Div (ex X)

0

11.12

Proof. The first statement follows by integration of the infinitesimal contributions 9 to curl ex X . The second statement, curl curl ex X = 2curl ex 1 X = h ex 1' X =

-ll- ex X , is immediate. The third statement follows from the div curl Y

identity

= 0 , or by considering the infinitesimal contributions to ex X .

The differential equation the statement that symmetric part

curl2 ex X

+ ex X = O

ex f = av f is harmonic, when

I

is the counterpart to

is a function. The

f

V5 X of the covariant derivative measures

the infinitesimal

strain, or distortion of the metric, of the flow generated by X . That is, if

Y and Y' are vector fields invariant by the flow of X , so that

VYX= VXYandVY,X=VXY',sothe

[X,Y]=[X,Y']=O,then

derivative of the dot product of Y and Y'

in the direction X

,

by the

Leibniz rule, is

X(Y'Y')

ava-Y = WV?

+

y.vxIv

+

VYX

Y

2(V;X'Y') The symnetric part of V

and a part with trace 0

VSX

can be further decomposed into its effect on volume

, =

E’;1 Trace(VsX)'I

5‘0

+ VX

Here, Irepresents the identity transformation (which has trace

Note that

trace

VSX

= trace VX

divergence X

2

Va iX

3

in dimension

ei

is an orthonormal basis, so for a. vector field of the form ex X , s

VoexX.

where

VS

3 .)

Eei]

ex X =

11.13 Now let us consider the analogous decomposition of the covariant derivative

VX of a. vector field on the

VaX

VX = Define linear maps

and

5

and

(or any surface). There is a. decompositicr

Riemann sphere

3

g(traceVXH .

+

s + V O X.

of the tangent space to itself by the formulas

3x0!)

[VYX

thr)

NIH {fo

- w i

+

i

V1120

for any vector Y . (On a general surface, 1 is interpreted as a

900

counter-

clockwise rotation of the tangent space of the surface.)

11.1.1-I». Progos1tion.

BX

and

3x

5X E invariant

=

i—‘(trace VX) I

=

% 30

= V

under conformal

localcoordinates.

,VaX

[(div X)I+ (curl X)iI]

x

cbfles _o_f

metric.

Notational remark. Any vector field on __J

+

m

.

be written

The derivative of f can be written df =

This can be re-expressed interms of dz = dx + idy rand as df = f dz + f-

z

..J

X =

where

-l

z = 2

f

z

f x

-

1 f

y

d'z'

dz

6 f(z)?z

fx dx = dx

+

-

. , 1h

fy dy idy

11.11:f-z

3f =

Then

fz dz

HIN

(rx +if) Y

3f i f; d;

and

61‘. Similarly,

linear parts of the real linear map

3X

=

f; d;

i

are the complex

ear and complex conjugate

BX

=

fz dz 38;

and

are the complex linear and conjugate linear parts of the map

dX=VX. Proof.

If L:

n: —-’ m

is any real linear map,

then

-]2-'-(L-i5L°i)

.

L

PIN

(L+i°L°i)

is clearly the decomposition into its complex linear and conjugate linear parts.

A complex linear map, in matrix form

L:

'2] ,_

is an expansion followed by a b , is a symmetric -a rotation, while a. conjugate Linear map in matrix form

[1:

1

map with trace 0 .

To see that

inY

=

iVxY

Ex

is invariant under conformal changes of metric, note that

and write

3x

Exm

without using the metric as

-%

32E

[vyx

iVflX}

+

VXY

[vyx

%[[Y,X] We can now derive a nice formula for

11.1.5.

Proposition,

m mm

Y a

+

iVx iY}

ivflx

+ i[iY,X]].

V5

ex X

T2: (H3)

and a_nl

C:L

vector field X on

s:

u.15

VgexX

=

73 —1, f

S2

1*(3x'(rm)') cwx

.

M

Proof. Clearly both sides are symmetric linear maps applied to Y

to show that the equation gives the right value for

VY ex X - Y .

, so

VYexX'Y

=

ex[Yu,X]

_— .3. 817

x (where ex i Y, = 0)

[exiYa,X] 'exiYm

- , X] . iYu [le

K.) Therefore ,

vx p.

pf:

w13

d

K:

i 8n

5.

O =

2

F4

the point

Y

I

, at

,1: F

and also

SP.

Y

[Yw,X]

it suffices

From 11.1.2 ,

we have

l

”7

avx

13.1 Chapter

13

Orb ifolds

As we have had occasion to see, it is often more effective to study the quotient manifold of a group acting freely and properly discontinuously on a

space rather than to limit one's image to the group action alone.

It is time

now to enlarge our vocabulary, so that we can work with the quotient spaces of groups acting properly discontinuously but not necessarily freely. In the first place, such quotient spaces will yield a technical device useful for showing the

existence of hyperbolic structures on many 3-manifolds.

In the second place, they

are often simpler than 3-manifolds tend to be, and hence they often give easy, graphic examples of phenomena involving

3—manifolds. Finally, they are beautiful

and interesting in their own right.

13.1. Some

example5 of

u notth

sac

s.

We begin our discussion with a few examples of quotient spaces of groups acting properly discontinuously on manifolds in order to get a taste of their

geometric flavor.

13.1.1. .3 ins}; mirror. Consider the action of

22

on

The quotient space is the half-space

183

by reflection in the y

a. person in this half-space is like all of

22

symmetry.

plane.

x Z O . Physically, one may imagine a mirror

placed on the y .. z “Ta-Jr]- of the half-space x

the

-z

ZO

IRS ,

. The scene as viewed by

with

scenery

invariant by

13.2

13.1.2. _A; m $11—02. Consider the group G generated by reflections in the planes

x = l in

123 .

G is the infinite dihedral group

quotient space is the slab O

SxSl .

D”

=

x =O

22 *Z2 .

and

The

Physically, this is related to two mirrors

on parallel walls, as commonly seen in a barber shop.

13.1.3.

A billiard M.

Let G be the group of isometries of the Euclidean plane generated by reflections in the four sides of a rectangle R . G

quotient space is R

. A

is isomorphic to

D”

x

D”

,

and the

physical model is a billiard table. A collection of balls

on a billiard table gives rise to an infinite collection of balls on

182 ,

G . (Each side of the billiard table should be one ball diameter

invariant by

larger than the corresponding side of R

so that the centers of the balls can take

any position in R . A ball may intersect its images in

o

’O

‘I

s

I

o

o

,3

\

9(‘

.

/

7 O

Ignoring spin, in order

to

of the images of y by G

13.l.1+. g rectanflar

R2.)

make ball x ,

hit bfll y

it suffices to aim it at any

(Unless some ball is in the way.)

pillow.

Let H be the subgroup of index 2 which preserves orientation in the group

13.3 G of the preceding example. A fundamental domain for H consists of two adjacent rectangles.

The quotient space is obtained by identifying the edges of the two

rectangles by reflection in the common edge. ” -e‘i

"

a.

Topologically, this quotient space is,a sphere, with four distinguished points or

1R2

singular points, which come from points in

The sphere inherits a Riemannian metric of O

’4-

points, and it has curvature

points

KPi

=

with non-trivial isotropy

( Z2 )

curvature in the complement of these

1r concentrated at each of the four

pi . In other words, a. neighborhood of each point pi is a cone, with cone

angle Tr

= 2.“.

- KPi

P;

0

m =-

Exercise. On any tetrahedron in

every geodesic is simple.

strips of paper.

fl

Rs

G

all of whose four sides are congruent,

This may be tested with a cardboard model and string or With

Explain.

- presently cmtallogranhic 3 - dimensional example to illustrate the geometry of quotient

13.1.5. g orientation Here is one more

out.

13.11spaces. Consider the

(m+%,t,n) a real parameter.

and

3

families of lines in

1

(n,m+-,t)

a

where

1&3

of the form

n and m

(t , n , m +

are integers and t

%) ,

is

They intersect a cube in the unit lattice as depicted.

Let G be the group generated by

180°

rotations about these lines.

It is not

hard to see that a fundamental domain is a trait cube. We may construct the quotient

space by making all identifications coming from non-trivial elements of G acting on the faces of the cube. This means that each face must be folded shut, like a book. doing this, we will keep track of the images of the axes, which form the singular

c

locus.

@

In

13.5 As you can see by studying the picture, the quotient space is

S3

locus consisting of three circles in the form of the Borromean rings.

with singular

S3

inherits

a Euclidean structure (or metric of zero curvature)

in the complement of these

rings, with a cone-type singularity with cone angle

11' along the rings.

In these examples, it was not hard to construct the quotient space from the In order to go in the opposite direction, we need to lmow not only

group action.

the quotient space, but also the singular locus and appropriate data concerning

the local behaviour of the group action above the singular locus.

13-2

Basic definitions

«-

An orbifold actions.

[

modulo finite group

Here is the formal definition: 0 consists of a Hausdorff space

with some additional structure.

sets

an

0 is a space locally modelled on

Ui

}

finite group homeomorphism

X0

is to have a. covering by a collection of open

closed under finite intersections. To each

Pi cpi

, an :

action of

Ui

3

vi

XO ,

ti

on an open subset

/ 1"i .

Whenever

Ui

Ui

iii C

is associated a

of

Uj ,

Ru

and a

there is to be

an injective hOmomcrphism

ij‘i

a

and an embedding

p l (U) consists of components of the form U / I", I" C I‘ . Let system. We may suppose that

_

0:.L

(i = l , 2) , and consider components of

be covering orbifolds

U / P1

for notational convenience we identify with

we can write the notation

and

726P

91,72 y

U / Pl ] . For

1" 7 n 1 l 7 1 and P

7‘1 1" 7 . 2 2 2

72 y _Of

course, f

272. Furthermore, for any

factors through

depends only on the cosets

71:72

7 e 1" , the maps

f71’72

U ‘,

are identical so only the product

really matters. Thus, the

invariant of

f

71,72 is

the double coset

in the fiber product of coverings

X1

and

f717’727

in particular,their images

7'1 1‘2

F 7 1. 1 2

and

71

,bytheformula

differ only by a group element acting on

71-72:L

which

Formally,

each pair of elanents

£71,72

In fact,

(U) ,

be more consistent to use

U/l"le/1"2

U—->

7 ,72

U / P2 .

and

U} . [It would

y e

,weobtainamap f

(Fl 7 1 y , F2

U / 7'1 1

Pl

U / F1 = [Fl y I Fl\ U instead of

pit-1

X2

e

"real."

Pl\F / 1‘2 .

of a. space X

,

(Similarly,

the components are

parametrized by the double cosets

‘rrle\11'lX / 7T1X2) .

U / F1

defined now to be the disjoint union, over

and

U/

over

l"

,is

7 representing double cosets

elements

U / rln

U / F2

7'l F27

U / r2 ,

.

I‘l\

I"

/

F2

The fiber product of

of the orbifolds

We have shown above how this canonically covers

via the map

fl,7 .

U / F1

and

This definition agrees with the usual definition

of fiber product in the complement of

£0 .

These locally defined patches easily

fit together to give a fiber product orbifold

spaces, a universal covering orbifold

.over some suitable set

0

U

01

3 O2 . As

in the case of

is obtained by taking the inverse limit

representing all isomorphism classes of orbifolds.

The universal cover

U

of an orbifold

O

is automatically a regular cover:

13. 11!» for any preimage

:9

5':

of the base point

* there is a deck transformation taking

3':

to

13.2.5.

The fundamental

Definition.

"1(0)

53mm

of an orbifold

o

is the

6.

group of deck transformations of the universal cover

The fundamental groups of orbifolds can be computed in much the same ways as fundamental groups of manifolds:

Trl(0)

in

terms

see

§

. Later we shall interpret

.

of loops on O

Here are two more definitions which are completely parallel to the definitions for manifolds .

13.2.6.

Definition.

_an

on

or

X0 X50

13.2.7. X

1

IR:

modulo finite groups and

When

80

g orbifold with boundary means

C X

a space locally modelled

modulo finite groups.

is a topological manifold, be careful not to confuse

5X0

with

a

Definition. A suborbifold

C’2

01

of an orbifold

le C an

locally modelled on

modulo

Thus, a triangle orbifold has seven distinct orbifolds, up to isotopy:

Dn

one

81

D...

and six mI's

02

means a subspace

finite groups.

"closed" one-dimensional

. i o1d The orb'f

An,m,p

with

seven suborbifolds .

DP

Note that each of the seven is the boundary of a suborbifold with boundary .

(defined in the obvious

way)

9

with universal cover D' .

sub-

13.15 13.3 T_W2-dimensional

orbifolds

To avoid technicalities, we shall work with differentiable orbifolds from DOW

on.

The nature‘of the singular locus of a differentiable orbifold may be

U =

Let

understood as follows.

There is a Riemannian metric on

U

obtained from any metric on

U

U

/ F

be any local coordinate system.

invariant by

F : such a. metric may be

by averaging under

1"

. For

any point

consider the exponential map, which gives a diffeomorphism from the

3':

in the tangent space at

to a small neighborhood of

map commutes with the action of the isotropy group of isomorphism between a neighborhood of the image of

an /

of the origin in the orbifold

U

8 ball

Since the exponential

3': , it in O

e

gives rise to an

,

and a neighborhood

l" is a finite subgroup of

On .

the orthogonal group

13.3.1.

I‘ , where

3':

3': .

5':

Pronosi ion.

The sin

ar locus _o_f

a: two-dimensional orbifold h_as_

these types o_f local models

(i)

km:

122 / 22 ,

where

22

ax_is.

(ii)

Elliptic points

of

order n :

m2 /

acts by reflection Eth_e y

zan ,

with

zzn

acting

BY

rotations.

(iii)

(logr reflectors _of o_rfl n dihedral group of order

ma / DI1 ,

where

2n , with presentation

Dn

is the

13.16 The generators a

at

correspond to reflections in lines meetigg

and b

angle w / n .

r1/

02 .

Proof. These are the only three types of finite subgroups of

It follows that the underlying space of a two dimensional orbifold is always a topological surface, possibly with boundary.

It is easy to enumerate all

2-dimensiona1 orbifolds, by enumerating surfaces, together with combinatorial

information which determines the orbifold structure. From a topological point of view, however, it is not completely trivial to deterndne which of these

orbifolds are good and which are had. ,.

“W

We shall classify 2-dimensional orbifolds from a geometric point of view. When

G is a group of real analytic diffeomorphisms of a real analytic manifold

X , then the elementary properties of (G , X) case of manifolds

(see

§ 3.5) . In particular D :

can be defined for a (G , X) of paths in O

,

- orbifolds are similar to the

U

-9

- orbifold

a developing map

X O . Since we do not yet have a notion

this requires a little explanation.

Let

{

Ui

} be a covering

of O by a collection of open sets, cloSed under intersections, modelled on

13.17

Ui / I‘i

,

wit

C X

isometries

Ui

Uj

Ui

D

C

U1

,

such that the inclusion maps

UO

. Choose a "base" chart

Ui C Uj

. When

UO

come from 3

Ui

C

1 (a simplicial path in the

is a chain of open sets

2 2n 1 skeleton of the nerve of [ Ui }) , then for each choice of isometries of

-

the form

U0

70¢i,0 1

,.

Ul

72%1’i2

Ui

one obtains an isometry of

a

in X

,

U.1211

“

U.12

(—

obtained by composing the transition

~

fimctions (which are globally defined on X). A covering space 0 of O

is defined

Ui

pri)»

can: where Cp is any isometry of obtained by the above construction. These are glued together by the obvious "inclusion" maps, ( q) ,

by the covering[ (q) ,

(pUi)

c_, ( '4’:l0

U3

"

.q)

-l

) whenever I)

73 .351'].

is of the form

for some

I" 7:163 The reader desiring a picture may construct a "foliation" of the space

[(x , y , g)

x e X ,y e

Xo , g

is the germ of a

G—map between neighborhoods

y] . Any leaf of this foliation gives a developing map.

of x and

13.3.2.

I

Proposition.

When

G i_s g

X , then evez_'y (G , X)

manifold

holonm

analfiic group 9f diffeomorphisms

- manifold E good.

o_f g

A developing map

homomorphism

are defined.

If

G

E2 Eoup 5g

closed gr metrically

gparticuier, i_f

x

isometries actig transitively g X , then

complete ,

E i_s emulate. (i.e., D

i_s simply connected , mg

6

0 i_s

_‘E 2 coverg‘ map) .

= X __9

”1(0) 22

13.18 discrete

suonup o_f G .

area:

§ 3.5.

see

Here is an eSample.

A2 3 6 3

2

has a Euclidean structure, as a

30° , 60° 90°

The developing map looks like this:

triangle.

I_

/.

/

_ / \/

..3/

I

,

,,

\/ «" /xx71\’ 4/ /I\ /\\>//\/:IX/1 A.‘

l

/,

/

l» .

‘

/

~

Here is a definition that will aid us in the geometric classification of

2— dimensional orbifolds.

13.3.3 . Definition. _— each open

cell

When an orbifold

is in the same

O has a cell-divison of

Xo

such that

stratum of the singular locus (i.e.! the 33°11? ESSOCJ-avf

to the interior points of a cell is constant), then the Euler number X(o) is

13.19 defined by the formula

Z

x(o)

(c.)

d'

(.1) m

1

(l/

[P(ciH),

C.

l

where

ci

ranges over cells and

associated to each cell.

IP(ci)I

is the order of the group

P(ci)

The Euler number is not always an integer.

The definition is concocted for the following reason. Define the number _o_f sheets of a cover to be the number of preimages of a non-singular point.

13.3.h Proposition . X(6) =

k X(o)

—>

o ise coverigg mwith

k

sheets, then

. verifiedtha-t

Proof : It is easily by the ratio

#

6

If

z

sheets =

the number of sheets of a cover can be computed

(IPxI

III—i”

/

:

i 3p(x) =x where

x is g point.

The formula

for the Euler number of

a cover follows immediately.

As an example, a triangle orbifold A

n1’I‘2’r‘3

1/2(Z(1/ni)-l).

+

D

NIH D

“3

a:

3

.1 2 g 2

+1

2n2

n2

has Euler number

13.20

A2 3 5

Thus,

9

3

has Euler number +

1/60 . Its

Se ,

universal cover is

with

deck transformations the group of symmetries of the dodecahedron. This group

x(h2 3 6) = o covered by $2

has order

120 = 2/(1/60) . On the other hand ,

)((A2,3 7)

= -l/8)+. These orbifolds cannot be

3

3

3

and

*

The general formula for the Euler number of an orbifold O with k corner

n:L

reflectors of orders

,..., nk

z

and

elliptic points of orders

m1

,..., 11113

is

13.3.u.

x(xo)

x(o) =

S x(Xo) ,

Note in particular that X(o)

X0

“3": 2(1 - l/ni) - 2(1 .' 1/mi) with equality iff O

is the surface

O-=mXo .

or if

If 0 is equipped with a metric coming from invariant Riemannian metrics

U , then

on the local models

one may easily derive the Gauss-Bonnet theorem,

f

13.3.5.

0

KdA

211' x(o) .

One way to prove this is by excising small neighborhoods of the singular locus, the usual Gauss-Bonnet theorem for manifolds with boundary.

and a

For 0

to have an elliptic, parabolic or hyperbolic structure, X(o) must be respectively positive, zero or negative.

2n

If 0 is elliptic or hyperbolic, then area

(0) '=

[x(o)] .

13.3.6.

Theoran .

A glLsed

E gmerbolic structure

A2 orbifold parabolic

0

2-dimensional orbifold

E5 2Mrs—rifle ,

i_fg E E gEd.

EEE hmrbolic

structurei_ff_ x(o) < 0, Egg

structure iff X(0)=O. An orbifold is

elliptic

Egg;

13.21 x(o) > o .. All bad,

elliptic

and

parabolic

orbifolds

£9.M below,

(nl ,..., bk; m1 ,..., mg) denotes _a_; orb ifold with elliptic points 9_f o_r—___ders 1:1:L ,..., nk aldcorner reflectors of orders m1,...,mz. M

where

not listed are

hmrbolic Bad

Elliptic

(

n

(n)

(“1 ’n2) I31< n2

The sphere

n)

a

(2 , 2 , n)

. 3) (2 , 3 . 1+) (2 , 3

S2

Parabolic

(2,3;6)

(2,h,h)

(3,3,3) (2,2,2,2)

(2 , 3 , 5)

($n’n)

(H11 , n2) n1 < n2

(3222,11) (; 2 , 3 , 3)

The disk

D2

(32.3.10

(53.3.3) (,2,2,2,2)

($2.3,5)

(2;2.2)

(ns)

(333)

(2;m)

(1H2)

(352) The projective plane

The torus The Klein bottle

K

The annulus A

The Moebius band M

2

JP

(;2:3,6) (,2,1+,1+)

(3.))

l3.a.a. 2 D(;LL,’+,’J.-)

n 5 q-n' —" The universal c.ve--_ng of spec.

and

2 SWAN-4')

generated by reflections in the faces of one of the tri

-

9

.e s of this ti__'1g 1* of F symmetfi”'1 I

.

'2 71'1 (D(-,’+,Lt,h)/ \

is

les. The full group of

2 - ”1(D(.,2,3,a))

is

O

This picture was drawn with a computer by Peter Oppenheimer.

~

wee; i‘1:3VI ,_

......l "9'."7‘ _ 3: 1"",

.

_.

..-

..———

.

_

13.22

M.

It is routine to list all orbifolds with non-negative Euler number,

as in the table. We have already indicated an easy, direct argument to show the orbifolds listed as bad are bad ; here is another. First, by passing

to covers, we only need consider the case that the Imderlving space is

$2 ,

and that if there are two elliptic points their orders are relatively prime. These orbifolds have Riemannian metrics of curvature bounded above zero,

which implies (by elementary Riemannian

geometry) that

must be compact. But the Euler number is either 1 is a rational nmber with numerator

any surface covering them

+i-

or

>2 .

Since no connected surface has an Euler

1— n1

+

«l— ,

n2

which

number greater than 2 , these orbifolds

must be bad. Question

What is the best pinching constant for Riemannian metrics on these

orb ifolds '2

All the orbifolds listed as elliptic and parabolic may be readily identified as the quotient of

$2

correspond to the

or

E2

modulo a discrete group.

The

17 parabolic

17 "wallpaper groups". The reader should unfold

orbifolds

these orbifold

13.23 for himself, to appreciate their beauty. Another pleasant exercise is to identify the orbifolds associated with some of Escher's prints. Hyperbolic structures can be found, and classified, for orbifolds with negative Euler characteristics by decomposing them into primitive pieces, in a

manner analogous

to our analysis of Teichmiiller space for a. surface

Given an orbifold

0

with

x(o) < 0 ,we may repeatedly cut it

(§ 5.3) .

along simple

closed curves and then "mirror" these curves (to remain in the class of closed

orbifolds) Lmtil we are left with pieces of the form below. (If the underlying surface is unoriented, then make the first cut so the result is oriented.)

n

EQZ

Z“a

13.2hThe orbifolds mP

2

case

A(n

,

. ) and D

3

A(2,2; )) and S(n1,n2,n3)

(D1 2 n2

3 )

(except the degenerate

have hyperbolic structures parameterized by the

lengths of their boundary components. Theproof is precisely analogous to the classific: tion of shapes of pants in

§ 5.3

(see § 2.6) .

"generalized triangles"

The orbifold

; one decomposes these orbifolds into two congruent

D? _

,..., m!

9

triangles",

for instance in the pattern above. hyperbolic structures (provided X

(

33+ )2-3 .

"generalized

0”,,

0"‘1

L

,

"'4

3

DM

that is

0

also can be decomposed into

One immediately sees that the orbifold has

< O)

parametrized by the lengths of the cuts;

(Special care must be taken when, say,

ml = m2

2

13.25

Then one of the cuts must

be omitted, and an edge length becomes a parameter.

In general any disjoint set of edges with ends on order 2 corner reflectors can be taken as positive real parameters, with extra parameters coming from cuts

is

not meeting these edges

a

Q4

:1. The annulus with more than one corner reflector on one boundary component be diesected, as below,

corner reflectors .

Dan

D
2 . D(2;2,2) is not

they are determined by their edges of length

and

However, it has a degenerate hyperbolic structure as an

rectangle, modulo a

rotation of order

A

2

—

infinitely thin

or, an interval.

k) This is consistent with the way in which it arises in considering hyperbolic

structures, in the dissection of D2(2;m1,...,mz)

'

One can out such an orbifold

along the perpendicular arc from the elliptic point to an edge, to obtain

D(2 ; 2 ,2 ’”1"“ ,mz).

. In the case of an annulus with only one corner reflector,

13.27

f

&

M

010nm)

note first that it is symmetric, since it can bedissected into an isosceles

"triangle".

now,

from a second.dissection,we see hyperbolic structures are

parametrized by the length of the boundary component without the reflector.

By the same argument,

D%n-

9

m) has a unique hyperbolic structure.

All these pieces can easily be reassembled to give a hyperbolic structure

on 0 . l

From the proof of

13.3.6

we derive

-

13-3-7 Corolla Ens Teichm'uller ems J(0) 21:21.1 orb ifold 0 113 X(O)

+t

Ln 20 ,m 20] .\-

+

[I [:0

O

.

e

Z€L(T”,

(1r

-e(e))+2rrx(szf.)

)

This may be deduced quickly by comparing the R metric with a metric which

R' in

R' (z/') is near 0 . In other words, the image F(A) is contained in the

interior of the polyhedron P C Z defined by the above inequalities. Since F(A) is an open set whose boundary is

this completes the proof of

5P , F(A) =

interior

(P) . Since 0 e int(P) ,

13.7.1. 13.7.1, 13.6.h,

'

13.6.5

Remarks. This proof was based on a practical algorithm for actually constructing

patterns of circles. The idea of the algorithm is to adjust, iteratively, the radii of the circles. A change of any single radius affects most strongly the curvature

at that vertex, so this process converges reasonably well.

13.61 The patterns of circles on surfaces of constant curvature, with singularities

at the centers of the circles, have a the inclusions

isom

(HZ)

C isom

(H3)

associated with such a surface S to S

x JR ,

3-dimensional and isom

interpretation. Because of

(E2) C. isom (H3) , there 3-manifold

a hyperbolic

Ms

,

homeomorphic

(the singularities of S) x IR

with cone type singularities along

Each circle on S determines a totally geodesic submanifold

(a "plane") in

These, together with the totally geodesic surface isotopic to S when

cut out a submanifold of

13.6.h

or

13.6.5

Ms

with finite volume

is

—

MS

.

S is hyperbolic,

it is an orbifold as in

but with singularities along arcs or half-lines running from

the top to the bottom.

13.7.h. Carol—1.5g. Theorems 13.6.u Bid 13.6.5 hold when 8 _i_gg mclidea: g hmrbolic urbifol , instead 23 _a_. surface. (TE orbifold 0 _i3 19 have 9311 sflities fig 13.6.h_&r_ 13.6.5, 'Dlus (singularities of s) x: g (singxflarities of s) x

[o

, w)

:)

Proof. Solve for pattern of circles on S in a. metric of constant curvature on

S

—

the underlying surface of S will have a Riemannian metric with cone type

singularities of curvature 2n (

%-

l) at elliptic points of S , and angles at

corner reflectors of S . An alternative proof is to find a surface of the

orbifold

covering space

S

6

, and

S

which is a finite covering space

find a. hyperbolic structure for the corresponding

of 0 . The existence of a hyperbolic structure for 0 follows

from the uniqueness of the hyperbolic structure on O thence the invariance by

deck transformations

of 6

over 0

.

13.62 compactificat ion $2; Egg Teichm‘iller 13.8. A geometric ___—___-

spaces

9:

orbifolds.

We will construct hyperbolic structures for a much greater variety of

orbifolds by studying the quasi-isometric deformation spaces of orbifolds with boundary whose underlying space is the

three-disk. In order to do this, we

need a description of the limiting behaviour of conformal structures on its boundary.

We shall focus on the case when the boundary is a disjoint union of

For this, the greatest

polygonal orbifolds. right

clarity is

attained by finding the

compactifications for these Teichmfiller spaces.

When M

is an orbifold,

M[5 m)

is defined to consist of points

3

8/2 about x

such that the ball of radius

x in M

has a finite fundamental group.

Equivalently, no loop through x of length < 8 has infinite order in

M

(0.6]

is defined similarly.

.

It does not, in general, contain a neighborhood .

(as in §5) that each

of the singular locus. With this definition, it follows

component of

M(O

axis, and its

fundamental group contains Z or Z EB Z

5

ni(M)

s] is covered by a horoball or a unifonn neighborhood of an with finite index.

In §5., we defined the geometric topology on sequqnces of hyperbolic 3manifolds of finite volume. For our present purpose, we want to modify this definition slightly. First, define a

hmrbolic

structure

dimensional orbifold O to be a complete hyperbolic

with Eggs; on

structure with

a

2-

finite volume

on the complement of some l-dimensional suborbifold, whose components are the

ngdeg. This includes

the

case when there are no nodes. A topology is defined

on the set of hyperbolic structures with nodes, up to diffeomorphisms isotopic to the identity on a given surface, by saying that

if there is a diffeomorphism of O

to M

Hanan)

is a

(e5) -

M1

and

M2

have

distance

5

a

[isotopic to the identity] whose restriction

quasi-isometry to

M9[e' — ,m)

. Here, 5' is some fixed,

13.63 small number.

M15.

The related topology on hyperbolic structures with nodes up

diffeomorphism on a given surface is always compact.

E

(Compare J¢rgensen's theorem,

5.12, and Mumford's theorem, 8.8.3.) This gives a beautiful compactification for the modular space

Earle and Marden

7 (M) / Diff(M)

,

which has been studied by Bers [

].

] and Abikoff [

[

],

What we shall do works

because a polygonal orbifold has a finite modular group.

For any two-dimensional orbifold O with X(o) < O,

the

Theorem . When P

EE

be the

(up to isotopy) on. O .

space of all hyperbolic structures with nodes

13.8.1.

let77(0)

n-gone]. orbifold ,

‘7](P) E homeomorphic t_o

Dn'3 , with interior j(P) . pt has 3 natural cell-structure open cells were 2y the E o_f nodes (pp 2 isotopy).

(closed) disk

with

,

Here are the three simplest examples.

J (P)

24mm EX If P

nodes

is a quadrilateral, then

77(P)

is

IR . There are two possible

looks like this:

“V S 24-4.

If there are two adjacent order 2 corner reflectors, the qualitative picture

must be modified appropriately. For instance,

’ 0"

When

1

1‘3—

P is a pentagon,

j(P)

is

R2 .

and the cell-structure is diagrammed below:

There are five possible nodes,

13.61;

ad 3

(.6

End

5,

P

WP)

When there is only one node, the pentagon is pinched into a quadrilateral and a triangle, so there is still one degree of freedom.

When P is a hexagon, there are

9

possible nodes.

Each single node pinches the hexagon into a pentagon and a triangle, or into two quadrilaterals, so its associated

division of

8D3

2-cell is a pentagon or a square. The cell

is diagrammed below:

13.65 (The zero and one dimensional cells are parameterized by the union of the nodes of the incident

Proof as in

2-cells.)

92 13.8.1. 5.12

that P

(0,51

and

It is easy to see that

8.8.3,

77(P)

for instance. In fact, choose 8 sufficiently small so

is always a disjoint union of regular neighborhoods of short arcs.

Given a sequence

{Pi} ,

of the components of

we can pass to a. subsequence so that the core

Pi(0,5]

[Pi]

arcs remain bounded in

[C21 ,..., Elk}

. The lengths of all such

(this follows from area considerations), so there

is a subsequence so that all lengths converge

[z(ai) I z(ai) Z

l-orbifolds

are constant. Extend this system of arcs to a

maximal system of disjoint geodesic arcs

converges in

is compact by familiar arguments,

- possibly to zero.

But any set of

0] defines a hyperbolic structure with nodes, so our sequence

'77 (P ) .

Furthermore, we have described a covering of diffeomorphic to quadrants, so

it

‘77 (P )

by neighborhoods

has the structure of a manifold with

corners. Change of coordinates is obviously differentiable. Each stratum consists of hyperbolic structures with a prescribed set of nodes, so it is diffeomorphic

to Euclidean space (this also follows directly from the nature of our local

coordinate systems. ) Theorem

13.8.1

follows from this information. Here is a little overproof.

An explicit homeomorphism to a disk can be constructed by observing that has a natural triangulation, which is dual to the cell structure of

356(P)

a 77(P)

.

This arises from the fact that any simple geodesic on P must be orthogonal to

the mirrors, so a geodesic lamination on P

is finite.

The simplices in

are measures on a maximal family of geodesic l-orbifolds.

*For definition, and other information, see

p.

8.58.

Wifi)

*

13.66 A projective structure for homeomorphism to a sphere

955 (P) - that

flafi (P) .

take the directional derivative at

convex hull, in the direction f .

132

to

IR is added to f . Thus,

To lift the map to measured laminations,

O of the bending measure for the top of the The global description of this map is that a

function f is associated to the measure which assigns to each edge e of the bending locus the change in slope of the intersection of the faces adjacent to

e with a plane

perpendicular to c

It is geometrically clear that we thus obtain a piecewise linear homeomorphism.

*

See remark

9.5.9-

13.67 e

W£(P)

==

le-3

O

The set of measures which assign a maximal value of l to an edge gives a

€06 (P)

realization of

*

hedron Q,

-

-

which is, by definition, the set of vectors

- Y=

supyeQ X

l

13-9- a geometric 52°11'05-

compactification

6

_

EV 3 such that

man defamation spaces a certain Kleinian

Let 0 be an orbifold with underlying space

520

X

The dual poly-

— is the boundary of a convex disk, combinatorially equal to 77(P).

This seems explicit enough for now.

and

lav-'3.

as a convex polyhedral sphere Q in

x0

D3

,

20

c

3133

a union of polygons.

We will use the terminology Kleinian s_tructure on 0 to mean a diffeomorphism of

O

to

. a.

Kleinian manifold

33 - LI. / 1" ,

where

I" is a Kleinian group.

In order to describe the ways in which Kleinian structures on O can degenerate, we will also define the notion of a

on

with nodes

O . The nodes are meant to represent the linu‘ting behaviour as some l—dimensional

suborbifold S becomes shorter and shorter, finally becoming parabolic.

shall see that this happens only when

We

S is isotopic in one or more ways to

the geometry depends on the set of suborbifolds on being pinched in the conformal geometry of

30

30

50

;

isotopic to S which are

. To take care of the various

possibilities, nodes are to be of one of these three types:

(a) An incompressible l-suborbifold of 30 . (b) An incompressible 2 ‘ dimensional suborbifold of O , with Euler characteristic zero and non-empty bomdary. In general, it would be one of these five:

.

13.68

mirrors

mirror

but for the orbifolds we are considering only the last two can occur.

P2}: x m , k > 2 where P21: is a of P2]: are to alternate being on 30 and and b could be subsumed under this case

(c) An orbifold T modelled on polygon with

El: sides.

in the interior of O

The sides

. (Cases a

by thickening them and regarding them as the

cases k = l and k = 2 .)

A Kleinian structure with nodes is now defined to be a meinian structure in the complement of a union of nodes of the above types, neighborhoods of the nodes in ‘oeing horoball neighborhoods of cusps in the Kleinian

structures. Of course, if 0

minus the nodes is not connected, each component is the quotient of a separate

Kleinian group

Let

(so our definition was not general enough for this case.)

'77 (0)

to homeomorphisms isotopic to

on

77(0) ,

the

up

identity. As for surfaces, we define a topology

by saying that two structures

a

and

if there is a homeomorphism between them which is an

KEEN)

intersected with the convex hull of

13.9.1.

Theorem . psi

I_f O has

,

denote the set of all Kleinian structures with nodes on O

K1 .

K2

have distance

5

E:

e6 - quasi-isometry on

pg 55 above with o irreducible and Bo incomnressible . one non-elemental-x Rainier: structure, then ‘fl (0) 1?. compact . E

conformal structure o_n

0

80

i_s_ c mtinuous , and it

77(0)

gives

2: homomorphism t_o_

.. 72 (50)

Note: The necessary and sufficient conditions for existence of a Kleinian

a disk,

C-

13.69 , or

structure will be given in theorem

13.6.1.

they can be deduced from Andreev's

We will use 13.9.1 to prove existence.

Proof. We will study the convex hulls of the Kleinian structures with nodes on O

. (When the

Kleinian structure is disconnected, this is the union

of convex hulls of the

pieces.)

13.9.2.

35 5 m

lam. There

H , pf p

upper bound for the

new structure with nodes pm

Proof

2; 13.9.2.

The bending

vol—“ale p; the

convex hull,

0 .

lamination

for

50

has a bounded number

Therefore, H is (geometrically) a polyhedron with a bounded

of components.

number of faces, each with a bounded number of sides. Hence the area of the boundary of the polyhedron is bounded.

Its volume is also bounded, in view of

the isoperimetric inequality, volume

foraset

s c:

Theorem

H3

(S)