Surfaces and Planar Discontinuous Groups (Lecture Notes in Mathematics, 835) 3540100245, 9783540100249

Citation preview

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

835

Heiner Zieschang

Elmar Vogt Hans-Dieter Coldewey Surfaces and Planar Discontinuous Groups

Revised and Expanded Translation Translated from the German by J. Stillwell

Springer-Verlag Berlin Heidelberg New York 1980

Authors Heiner Zieschang Universit~t Bochum Institut flit Mathematik Universit~tsstr. 150 4630 Bochum 1 Federal Republic of Germany Elmar Vogt Freie Universit~t Berlin Institut f~Jr Mathematik I H~ittenweg 9 1000 Berlin 33 Federal Republic of Germany Hans-Dieter Coldewey Allescherstr. 40b 8000 M0nchen 71 Federal Republic of Germany Revised and expanded translation of: H. Zieschang/E. Vogt/H.-D. Coldewey, Fl~chen und ebene diskontinuierliche Gruppen (Lecture Notes in Mathematics, vol. 122) published by Springer-Verlag Berlin-Heidelberg-New York, 1970

AMS Subject Classifications (1980): 20 Exx, 20 Fxx, 30 F35, 32 G 15, 51M10, 57 Mxx ISBN 3-540-10024-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10024-5 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

In memori~7 Kurt R e i d e m e ~ t e r

Introduction For the two-dimensional manifolds, the surfaces, the classical topological problems - classification and Hmuptver~atun S - have lonS been solved~ and more

delicate questions can be investigated. However, the most interesting side of sur-

face theory is not the topological, but the analytic. Results of the complex ana-

lytic theory, e.g., are often purely topo]osical~ but their proofs ape not, usin S

deep theorems of function theory. This results from a natural and close connection with discontinuous groups of motions in the non-euclidean or euclidean plane. The following lectures deal in the fim'st place with cori~inatorial topological

theorems on surfaces and planar discontinuous gn~oups{ thus we have adopted the concept of the book "EinfL~nruns in die kombinatorische Topologie" by K. Reidemeister.

Admittedly, in chapters 1-5 we have not kept strictly to the combinatorial conception~ but have changed to a~nother category where this seems converient. ~(v i), i = 1,...,m. Exercises:

E 1.13-17

[]

13 1,8

GEOMETRIC

INTERPRETATION

OF THE

NIELSEN

PROPERTY

In this section we give the geometric interpretation of the Nielsen property for generators of a subgroup from [Reidemeister-Brm~dis 1959]. Let S be the fundmmental group of a graph C wlnichhas only one vertex and let

U be the covering complex for the subgroup C. We take the free generators for S to

be those belonging to C. By a minimal s~nning tree with respect to P' of U we mean

a spanning tree which, for each point Q' of U, contains a paJ~ of m i n ~ a l length cormecting P' and Q'. As we saw in the proof of Theorem 1.3.3 e a ~

graph possesses

a ~/nimal spanning tree. The proof of Theorem 1.3.5 shows how we find free generators for U. If one determines the generators from a minimal spanning r which have a half in the tree already constructed.

stage one of the generators

If at this

still has neither half in the tree constructed so far,

we add one half to the tree and proceed as before. As above, one may convince oneself that each step results in a tree~ so that we finally arrive at a spa~ming tree B whid~ contains each vi, i : 1,...,m except for one s e c a n t . direction,

If we choose the right

the generating system relative to B for the fumdamental

b~sepoint P' corresponds

However,

group of U with

to the Nielsen generators Vl,...,v m and B is a minimal tree. []

if one starts from a minimal spanning tree, takes generators,

and again

constructs a minimal spanning tree, then one obtains a possibly different tree•

1,9

AUTOMORPHISMS

OF

A

FREE

GROUP

OF

FINITE

RANK

Let S be a free group of finite rank with free generators

Sl,S2,...,s n. If ~ is

an automorphism of S, then aSl~...,as n is again a system of generators sely, if V l , . .•, v n is a system of generators

of S. Conver-

of S, then s I• ~ v.l defines an auton~r-

phism. Now we can convert the s i into the v i by Nielsen processes

and obtain:

1.9.1 1%.eorem. Each automorphism of a free group S with free generators Sl.S 2 .... ,sn

is a product of the following automorphisms [Nielsen (a) s 1 ~

(b)

SlS2, s i ~

si

1919, 1924]:

i > 1 i > 1

s I ~ s~ 1, si ~ s i

(c) Permutation of generators.

J.H.C. Whitehead has given a constructive lowing problem:

Given elements w I , w 2 6 S, is there an autoKorghism of S which sends

w I to w2? J.H.C. ~ i t ~ e a d 1.9.2

process for the solution of the fol-

s s s

] ~j

~j

s j

~s

uses the following system of automorphisms: l~j_ 2, show that an endo~nrphism 6

with B(sat b) = sat b is an automorphism.

b) If an automorphism satisfies ~(sat b) = sa't bT then a' = ±a, b' = ±b

or a T = ±b, b' = ±a. (~nis exercise is difficu£t, cf. [Lyndon 1959]~ [Zieschang 1962]. )

2,

2-DIMENSIONAL COMPLEXES AND COMBINATORIAL PRESENTATIONS

OF GROUPS

Each group is a factor group of a free group, so in addition to a system of ge-

nerators we need a description of the kernel. ~lhis leads I:o the notion of a relation

and presentation of a group. In the following we consider g~oup presentations arid

their connection with 2-dimensional complexes. We generalize the methods of the first chapter to prove some subgroup theorems using coverings of these complexes.

2.1

TIETZE'S THEOREM

2.1.1 Definition. Let (S.).

be a system of sy~)ols and let (R.(S)). be a system ] 3@ J of words in the S i and STi 1. We say that a group G has the (combinatorial) presenta! l ~ I

tion

when

(a) there are elements s i 6 @, corresponding to the Si, which generate @, (b) for all j the relations R.(s) = 1 are satisfied in @ and 3 (e) for each system (vi) i6 I of elements of a p u p a homomorphism

H with Rj(v) = 1, s i - v i defines

G ÷ H.

We then write 0 = and say that O is given by the generators

(Si)i 6 1 an@ the defining relations (R.). 3 3 6 J" Mostly we consider finite or countable systems and write

or or similar. For the free ~ o u p with

generators $1,... we write . This manner of speaking is not strictly correct, since tk~e elements $1,$2...

are not elements of G, but it is often convenient, in order to avoid continual changes of notation, to treat symbols, words or gYoup elements interohangeably. In all cases it will bk clear from the context which meaning is intended. Later we shall

also write G = where the s. ~ e i

elements of @. We obviously have the

2.1.2 Uniqueness theorem. Two groups having the same presentation are isomorphic. Let S be the free group on the generators Sl,S2,... and R the smallest

20

normal subgroup of S which cor~ains the elements RI(~), R2(~) , .... The group S/R is

then generated by the elements si = Rsi, and Ri(s) : 1 for all i.

~ ~

If a group H contains elements Vl,V2~... such that R.(v) : 1 for all i, then vi defines

l

a h o m o m o r ~ h i s m a : S ÷ H w h i c h maps R o n t o t h e

a homomorphism a: S/R ÷ H, which maps s. : Rs. l

i

1 o f H. Thus a i n d u c e s

onto v.: i

2.1.3 Existence theorem. For each system ($1,$2,...; RI(S), R2(S), ...) there is a

group G : . 2.1.4 Remark. We see that any further relation of G~ regarded as an element R(s) of

S, lies in R. We call all words associated with elements of R consequence relations of the Ri(S). They are products of conjugates of the Ri(S) and their inverses.

It is clear that a group G can have different presentations. We now give pro-

cesses (2.1.5) - (2.1.87 by means of which one can pass from one presentation to all others. Let G : " 2.1.5 Addition of new generators U i and as ~ n y

W.l is a word in the S.. Z

2.1.6 The operation inverse to

new relations ~ . ( S , U i) = UiW~ 1 where l

2.1.5 .

2.1.7 Addition of consequence relations. Among these we also admit trivial relations,

i.e. words q u a l to 1 in the free group. 2.1.8 The operation inverse to

2.1.7 .

These operations are r~med after Tietze, who also proves 2.1.9 Theorem. Two presentations

and

define isomorphic groups if and only if presentation

0 boundaries, when F is non-orientable of genus k with r > 0 boundaries,

~2×Z k-1

when F is non orientable of genus k and closed,

Z

when F is closed an orientable

0

otherwise .

D

3.2.11 Notation. Tne following surfaces are the simplest ones and have special names. Name

(2 )-sphere

torus

(o) +

+

projective plane Klein bottle disk

a~mulus

Moebius strip

+

+

(b)

(

0

0

1

0

1 2

0

(f) 1 2 3

0

4

2

6

I

1

5 7

73 where (o) gives the orientability character (+ for orientable)~

(g) is the genus,

(b) the r~mmber of bo'~ndary curves and (f) the number in t]~e following figure

(1)

, I t ......

(S) l

(2)

(6)

(4) Exercises: E 3.1- 12.

3,3

KNESER'S FORMULA We introduce the notion of the degree of a mapping ar~ prove that a mapping of

degree c between sur~faces of (positive) genera g

and g' satisfies the inequality

g' - 1 ~ tel (g - 1) Following [Seifert 1937] we then deduce a short proof ot ~le Dehn-Nielsen theorem. In this section we admit surfaces which are not connected.

somewhat

the proof of Theorem 3.3.3.

This facilitmtes

Given a surface mapping f: F' ÷ F (see 2.7) we can use subdivision of faces of

F' to reach the situation where, in case 2.7.4 (b), f(~¢') traverses the boundary

of ~¢ exactly once, possibly with spurs, so in the future we shall assume this is the case. The coverings introduced in 2.4 are dimension-preserving

surface mappings

which satisfy from the outset the condition we have just achieved. images of stars of F' traverse stars of F exactly once.

3.3.1 Definition. A branched covering is a dimension-preserving

In addition, the

surface mapping with

74 the p r o ~ r t y

that the bouuldary paths of two faces of F' mapped onto [he same boun-

damy pat]~ of F have no edge in common. However,

stars in F' can m~ltip~]y cover stars

of F. 1]~e multiplicit-y of the covering is called

3.3.2 Definition. orientation:

the branching number of the vertex.

Let F and F' be c l o s e d o r i e n t a b l e

arid l e t

surfaces with a fixed choice of

F b e co~%nected. The d i f f e r e n c e bet%~een t h e number o f p o s i t i v e

f a c e s o f F' and n e g a t i v e f a c e s o f F' mapped by f t o p o s i t i v e f a c e s o f F does n o t depend on t h e c h o i c e o f f a c e o f F ( t h e p r o o f i s E x e r c i s e a t t h e end o f t h e c h a p ter)

and i s c a l l e d t h e d e g r e e o f t h e mapping f . I n [ F ~ e s e r 1930] i t

i s proved

If F and F' are closed orientable surfaces of genera g and g' respec-

3.3.3 ~ d n e o ~ .

tively (gT ~ 1) and if f is a surface mapping of F' onto F of degree c, then (g' - 1) ~ Icl(g - 1).

Proof.

Since g' > 1 we can also assume g >_ 1~ since otherwise the assertion is tri-

vial. I%~; neither F nor F' is a sphere, and we may further assume that F' contains no spheI~s

(rem~nber that we do not ass~me F' to be coraected).

Euler characteristics

that n' -< Icl

• n.

of F' and F respectively,

~en

Let n' and n be t/he

the ~ssertion of the theorem is

The following lemma, which we shall not prove with the means at our' disposal, justifies

the assumption d~at F' contains no spheres.

3.3.4 Lem~¢. A mapping of the sphere on to a surface of higher genus has degree O. ~} 3.3.5

If f: F' ~ F is a covering

(ur~ranched)

then over each geometric face geome-

tric edge ~mnd vertex of F @~ere are exactly c geometric faces, geometric edges and points respectively,

so that n' = Icl • n. If f is branched,

then again there are Icl

geometric faces and geometric edges over geometric faces and geometric edges of F, but at most

Icl

vertices over a vertex, so that n' ~ Jcl • n again holds.

to p_rove the Caeorem we now distinguish 3.3.6

In order

several cases.

f never lowers dimension. By means of subdivision we can first reach a situation where all faces of F are

1)Proof of the lemma: The universal covering of the surface F is the plane and therefore contractible. Consequently ~2(F) = 0 and each image of S 2 is contractible. The mapping degree is preserved under deformations. The degree of the constant mapping is 0 and the lemma follows.

75 triangles, no edge appears twice in the bou~dar% ~ of a tmiangle and no edge is closed, and we lift this subdivision to F'. (f is then a s]mplicial mapping. ) A segment a of F' is called a

~fold when

the two triangles having a as a boundary edge are mapped on-

Q

to the stone triangle of F.

When folds are present we alter F' in such a way ~ a t

served, but the Euler characteristic

the mapping degree is pre-

increases or the number of faces decreases. Af-

ter finitely many applications of this process all folds will have disappeared and a (branched) covering will remain. T h ~

the d~eorem will be established in this case.

%~nere are three t%qoes of fold.

(a) The two triangles which are mapped onto the stone one have only an edge RQ in common

P1

P2

We cut the two triangles from F' along the closed path PIQP2PP1 and stitch PP1 to PP2 and QP1 to QP2 in the surface which remains. Since the two triangles are mapped

with different orientations,

the mapping degree remains Ik~e same. The Euler cklarac-

teristic is unaltered and the new complex is also a closed surface. (b) The two triangles have two edges in corm~on.

R

We cut the cone from F' along ala 2 and stitch together a I and ao I in the remaining complex. Again neither the Euler characteristic nor mapping degree changes, and the complex remains a closed surface. (c) %~ne two triangles have an edge and the opposite vertex in co~tmon. (~qere are no

further cases, for if the two triangles have all edges in common they constitute a sphere )

76

T

R

a2

-i We tlnen cut F' along 01a 2 and stitch a I to a 2 . As a result, tkle Euler characteristic is raised by 2 and the mapping degree is unaltered. If Ola 2 separates the surface F' then the two new components carmot both be spheres, otherwise the component previously containing the triangles would have been a sphere a£ready. Thus we obtain from F' a new system of surfaces, possibly requiring removal of one sphere, which lowers tff-~echal~acteristic by 2. Aitogetiqer, ti~.eEuler characteristic of F ~ is raised if it is altered at all. After a finite number of such steps we obtain a branched covering, and the theorem follows from 3.3.5. 3.3.7

Let o' be a segment of F' which is mapped on to a point. If a' has two distinct

endpoints then we contract them into or-e (Process 3.1.6 (b)). If the two endpoints are identical then we cut along 0' and contract the resulting botmdary curves to a point. This raises the Euler characteristic by 2. If a' separates the surface F' then two components again cannot both be spheres. If one of the n~{ components is a sphere we leave it out. This process therefore does not dininish the Euler characteristic but it preserves the mapping degree. 3.3.8

If a face is mapped onto a vertex, then its boundary edges are mapped onto a

point and 3.3.7 mmy be applied. Finally, it remains only to treat the case where a face ~' is mapped on to an edge a, but no edge is mapped onto a point. Let _+1 ~ ' : a~ ... o' Then d~e a[ are mapped on to a and there is a subpath a'o' of n" z i i+1 ~ ' with image aso-s(s = + 1). We subdivide ~' with respect to 0'0' i i+1

77

J

/

/

Q

P and contract the initial p i n t

Q of a! and the final p i n t P of o' together. After l i+l finitely many such steps tile face which is mapped onto an edge has exactly two edges in the boundary, which are Kmpped onto the edge ~ and its inverse. We then cut F'

along ~ne two bo~nda~- edges, leave the face out~ and stitch up again. Since this process again does not alter the Euler ~haracteristic, and no face relevant to the computation of the mapping degree is involved, we are able to assume that f does not lower the d k n e n s i o n a n d the theorem follows from 3.3.6.

The rest of this section is not needed for the later text. We now extract some useful results from the proof of Kneser's formula. Unfortunately, it would be rather complicated and artificial to formulate them in te~ms of the combinatorial theory we have developed so far (it will be done somewhat in chapter 5), but this can easily be done using basic notions and results of algebraic

topology.

3.3.9 Corokl ~ .

Let F' and F be closed orientable surfaces of genus g' and g resp.,

and let f: F ' ÷ F be a mapping of degree c t o. If g' - 1 = Icl (g - 1)

then

f can be continuously deformed into a covering mapp£ng p: F' + F.

Proof (a bit sketchy). For g = o it follows ~lat Icl = 1 and g' = o. It is known that a mapping S 2 -~ S 2 of

degree ± 1 is ho~ntopic to a homeomor~hism, see E 3.29.

Now let us assume g'~ g >_ 1. In proving ~kneser's formula we altered the mapping f (and p s s i b l y F') without c~anging the degree. If we replaced F' by another surface the Euler characteristic went up. As we now ass~ne that equality holds, F' can never be changed substantially. More precisely: the p~oeesses 3.3.6 (a) and (b) can be replaced by a homotopic deformation of F such that the new mapping maps the two triangles to an edge. For process 3.3.6 (c) the curve ~1~2 bounds a d~sk and we can deform f so that this disk is mapped onto f(sl ) . For 3.3.7 the only possibilit~y is the appearance of a sphere and again we can deform f continuously so that the disk is

78

mapped to ~n edge. 3.3.8 describes a procedure for hap~dling mappings whiGh lower tie dimension (translate the process into the language of homntopic deformations). Hence we miy assume that f never lowers dLmension and has no folds, i.e. f is a branched covering. Since proper branching implies proper inequality in Kneser's formula, it follows that f is a covering.

[]

By si~dlar considerations one can prove the following result: 3.3.10 Proposition. Let F' and F be as before and let f: F' ÷ F

be a mapping of de-

gree O. Fix some complex C 2 in F. Then f can be deformed continuously into a mapping which maps F into the 1-skeleton C 1 of C 2.

From 3.3.9 we can deduce the following important theorem 3.3.11 of Nielsen. In Chapter 5 we give another proof of it which depends on£y on the material we have developed in these notes and which can be generalized to cases which cannot be covered

by tihe present method.

3.3.11 Theorem ([Nielsen 1927]). Let F be a closed orientable surface,

F the funda-

mental group of F and a: F ÷ F automorphism. Then there exists a homeomorphism ~: F ÷ F which induces ~, i.e. t# = a. Proof. Let Tl,~l,...,Tg~ng be a canonical system of curves on F, see 3.2.6 with r = 0 and genus g. Then their homotopy classes tl,Ul,...,tg,Ug generate f. The curve g [~i,~i ] bounds a disk D on F. We select curves ~l,~l,...,~g,~g for i=1 ~(tl),~(Ul) , . ..,~(tg),~(Ug) and define a map ~i ÷ Ti,~ i'

÷ u i' which maps the basepoint g to itself and hmndles the orientation correctly. Then the curve ~ [Ti,~ i ] is nulli=1

homotopic, so the mapping defined on the CLUX~Jes "t~l,...~

can be extended to a map-

ping }': D ÷ F. Since %' maps equivalent points of ~i ando .r~.1 (or ~i and

~1) to the

same point, ~ factors to a mapping %": F ÷ F which induces ~. Let c be the degree of ~". Y~neser's formula 3.3.3 now becomes g - 1 ~ Icl (g - 1) Hence either g = 1 or c equals O or _+ 1. If the degree is 0 the image of F can be deformed onto the 1-skeleton, hence its f,~nd~mental group is free. Since ~ is an automorphism, F must be free, which is not true (Proof, see E 2.4). If c : _+ 1 the statement 3.3.11 is a consequence of 3.3.9. ~ e

case g = 1 needs other considerations.

?9 One way is to use the automorphisms of Z e %, which are well known. We give it as exercise E 3.19.

Exercises: E 3.13-19.

3,4

C O V E R I N G S OF SURFACES As we have seen in section 3.2 surfaces are characterized by their Euler charac-

teristic, the ~ b e r

of bo~ndary components arid tlne orientabiiitybehavior.

Wewili

now give necessary and sufficient conditions in terms of these invariants for the existence of coverings between two surfaces with a given order. 3.4.1 Notation. In the following,F' and F denote connected surfaces which ean be presented by finite complexes. By X, g, r we denote the Euler characteristic, the genus and the number of boundary components of F~ similarly

X'~ g'~ r' for F'.

Our aim is the following theorem: 3.4.2 Theorem. Let F' and F be as in 3.4.1 and c ~ 1, c 6 Z. Then F' is a c-fold co-

vering of F if and only if X' = cx and one of the following conditions is ~%lfilled: (a) F' and F are both orientable and r S r' S cr. (b) F' and F are both non-orientable, (c) F' is orientable,

r ~ r' S cr and r ~ ~ cr mod 2.

F is non-orientable and 2r ~ r' ~ or and 21c.

~qe proof will be finished in 3.4.12. We follow

[HeLmes-St~er

1978] and look

for e ~ e d d i n g s of the fundamental group of F' in that of F with index c. ~nis approach is motivated by the facts we have proved in 2.5 and our treatment will give a good exercise in that

theory. We have to consider several cases. ~ne first case

3.4.3 is given as exercise E 3. 3.4.3 Case. F and F' are closed surfaces. Then the statemen% of 3.4.2 is true. 3.4.4 If p: F' ÷ F is an c-fold covering and x

a vertex of F, then the fundamental o group W of F acts on the set p-l(x o) of vertices of F' over Xo. ~ i s is a consequence of the unique path-lift~mg property for coverings, see the proof of 1.5.2. Choosing ~n enumeration of the vertices of p-l(x o) by 1,2,..0 ,c, we adjoin to each ele-

ment of W a pe~tutation from S c and obtain a homomorphism f: ~ ÷ S c, a so-called representation of {'J in the symmetric group S c . Here we use the fol:]owing notation: if x, y 6 S then xy(i) = x(y(i)) for 1 _< i ~ c. As F' is connected the image f(~) acts c transitively on {1,...~c}. If ~ o c-fold coverings over F are isomorphic then their representations in S c differ only by an isomorphism of Sc, in this case they are cal-

80

led equivalent. On the other hand, to each transitive representation of [J in S c there corresponds the subgroup of index c of elements which are mapped to permutations that fix 1. To this subgroup corresponds a covering of F, as we have seen in 2.5. If two representations are equivalent then the coverings are isomorphic. ~ u s

we have:

There is a 1-l-correspo~denae between c-fold (connected) coverings of F and equivalent transitive representations of W in Sc. (This assertion is true for all finite coverings between sufficiently nice topological spaces, see [Heimes-St6cker 1978] or books on algebraic topology.) Let y be a simple closed curve in F. Then y defines a set of elements in W in the following way: connect the basepoint x with the initial vertex of y by a path w. -1 o Then vy~ defines an element g 6 W. Zhis element is well defined up to conjugation. Homotopic curves define the smme conjugacy classes. I~l

If ~ C S then we denote by c the number of elements in the orbit space {1,...,c}/ where C S c is the

subgroup generated by ~. From -Lhe definitions we obtain directly: 8.4.5 Le~ma. With the r~tationfrom above, p-l(y) consists of If(g) l simple closed

curoes. [] Let us notice that g depends on y and ~, bu< f(g) only on ¥. By easy calculations we obtain the following from the equations ×' = 2 - r ' -2g' a{d × = 2 - k - 2 g : 3.4.6 Lemmm. If F ~ and F are orientable surTaces with b o u ~ r i e s r ~ r' ~ cr

r' ~ cr mod 2 and r ~ r' ~ [c(r-2) + 2

3.4.7

such that

and ×' = c× then

L

cr

if ~ = 0

if g > O.

[]

For orientable surfaces theorem 3.4.1 is a consequence of the following proposition. 3.4.8 Proposition. Let F be an orientable surface with boundary and let r' and c be

integers such that 3.4.7 holds. Then there exists a c-fold covering over F where the cover space has r ~ boundary components. We use 3.4.8 to prove 3.4.1 for the case where both surfaces are orientable and have boundaries. As noted above, from the assumptions r < r' < cr and ×' = c× we get

81

condition 3.4.7. By 3.4.6, F is c-fold covered by a surface F " w i t h r' boundary components. The Euler characteristic of F" equals c×~ hence, F" and F' have the same Euler characteristic and the same number of boundary components. As both are orientable they are homeomorphic by 3.2.8. Thus F' is a c-fold covering of F.

Proof of 3.4.8. The fundamental group of F has the presentation W :

r W(r,g) : .

We look for a representation of W in S c wher~ some generator is mapped to the cycle (1,2,...,c). Then the corresponding curve in F is covered by or2y one cur~'e, hence, the covering surface is connected. It is orientable as F is. Since r ~ 1 the group W is ~ne free group with free generators s2,...,Sr,tl,Ul,...,tg,Ug.

Thus, to construct a representation f: W ÷ S c, it suffi-

ces to give the images of these generators. We use the following abbreviation:

Zr : If(sl)l + "'" + [f(Sr)[ and rma x

We say that f realizes a given integer r'

jc(r-2)

+ 2

< cr if f: W ÷ S

if g = 0, if g > 0.

C

is a transitive representa-

tion such that Z r = r'. We prove the statement by induction and start with the induction step: Let g ~ 2 and suppose that fo: W(k~g-1) ~ S c realizes r'. ~nen we 'extend' f to W(r,g) by f(t ) : f(u ) : 1. Now f: W(r,g) - S realizes r', too. Therefore we g g c ray assume g ~ 1. Let r _> 4 or r = 3, g = 1, and suppose that fo: W(r-2,g) - S c realizes r'.o Extend fo to W(r,g) by defining f(Sr_ 1) = (1,...,m) and f(s r) = f(Sr_l )-1, where 1 S m S c. Since this does not change the f(si) , 1 ~ i S r-2, we get E r = Zr_ 2 + If(Sr_l)I + If(Sr) l = r'o + 2 ( c - m + 1 ) . If 3.4.8 is true for W(r-2,g), then r' may be any integer satisfying r' - c(r-2) rood 2 O

O

and r - 2 _< r' _< (r-2) . Therefore r' = r' + 2 ( c - m + 1 ) may be any integer such o max o that r' -= er m~d 2 and r < r' < r . Hence 3.4.8 is t~ae for W(r,g), arid besides max

g _< 1 we may assume that r _< 3, even r I there exists forevery c > 1 and every r' with r ~ r' _< cr a nonorientable surface F' with Euler characteristic ×' = c× and with r' boundary components; but F' is a c-fold covering of F only if in addition the congruence r' = cr rood 2 is satisfied. (In the orientable case this co~rruence is a consequence of ×' = c×.)

84 The problem of this section was first considered in [Massey 1974] and was there

solved by a geometric construction in the case where both surfaces are orientable.

The general theorem is from [Heimes-St~cker 1978]; our proof follows theirs. Parts of the new results of the paper [Heimes-St6cker 1978] were independendly

obtained

by 0. Fajuyigbe (Benin Cit%~, Nigeria).

3,5

T Y P E S OF SI M P L E C L O S E D

C U R V E S ON S U R F A C E S

We shall find an enumeration of the types of simple closed paths on surfaces.

First of all we give a process for deriving the homotopy class (path class) of a closed curve relative to a canonical system of generators for the fundamental ~roup. 3.5.1 Definition. We already know what a canonical curve system E = {~1...,~m,~1,~1, .... ~g,Vg) or {~l,...,~m,Vl,...,v S} of an orientable or non-orien-

table F of genus g with m boundaries is. Dual to Z, there is a system of simple curves

E* = {o~ ..... dm~ '~1~* '~1"~ ''" • 'T g"~ following properties :

~}~

or

{~1 .... '°'~m 'Vl{: '"''v~'}o with the

(a) All curves have a comrmn initial point P{~ and meet nowhere else. is not closed, and it connects P* to the ith boundary curve 0 i o f F. m g (e) F is decomposed by E* into a disc with the boundary H a.~ ~.~-1 ~ [T~ , ( ~ i:1 i Pi i j=l

(b) ~

or

m

~ a i Piai i=1

vI

...v g

)-1]

respectively.

(d) The star about P* has the form ~'"

c~ ,...,c m ,v1

,v~ -I .... ,Vg ,Vg -1

or

respectively when one considers only the

curves of E*.

(e) A curve n~

of ~' meets the corresponding curve n i of Z exactly once, and no

other curve of Z.

A system Z ~ which satisfies 3.5.1 (a) - (e) is called a canonical dissection of F. 3.5.2 Definition. We shall now define what it means for a curve ~ to positively in-

tersect a curve n~

of Z* in a point Pi" For this purpose we surround n i

by a nap-

row strip. For curves a'~m this strip is a "rectangle", for curves ~.~l,~ie a cylinder, and for curves v9m a M~bius strip. If the component of ~ in the strip, which passes through Pi' is a segment which runs from one side to the other, then in the

first two cases we can say whether 9k~e segment crosses the strip in the same direc-

85 tion as q i. If the strip is a M6bius strip then we cut it along an arc through Pe which does not meet ni~ so that it becomes a rectangle and we can a~sain decide whether the segment crosses the rectangle in the same direction as n i . This direction

is called positive, and the opposite direction is called negative. (In the latter case we are proceeding arbitrarily.

~he opposite convention will indeed change the

description defined below, however only defi~img relations come into play. ) 3.5.3 Defi~ition. We represent the curves n

,... ,nn

of the canonical dissection by

symbols H1,...,H n. Let ~ be a closed curve which does not pass through P~. We asso-

Cl ciate w with the word H 1

• ..

He£ e~ in the H i when ~ crosses

e.-

,

~-

qal "'',qa~

successively,

and ~i is + 1 or - 1 according as ~ intersects n . positively or negatively. Here we must assume that ~ meets each n I

l

properly at a point of intersection,

merely touch it. We call the word H i ... H ~

and does not

the description of ~. The curves n i ha-

ve the descriptions Hi, and a sinp!e closed curve which runs once around P* and con~.

-c.

tains no part H i iH i m has the description ~ ( H ) .

Conversely it is easy to convince

oneself that a simple closed curve with the description [ (H) bounds a dis~ around the point P~.

Naturally a cu~:e has different descriptions,

relative to different canonical

dissections. We shall give a distinguished description to each simple curve. For

this purpose we attach no significa~ce to the initial point of the curve - thus we

regard descriptions as cyclic words. Instead of changing the description when we go to a new canonical dissection, we can retain the old dissection and carry the curve into another by a homeomorphism H. 7his curve has the same description relative to

Z* as the old curve has relative to the dissection H-l(~f~ ), and for each canonical dissection Z*' there is a homeomorphism which maps E~ onto Z~'.

3.5.4 Theorem (Enumeration of types of simple curves). Let ~ be a simple closed curve

in the interior of F which does not bound a disk. Then there is a canonical dissection of F, relative to which ~ has one of the following descriptions. A.

F is orientable. If ~ does not separate, then the description is

3.5.5

TI~

if ~ does separate, then 3.5.6

Sq+I...S m

k ~ [Ti,Ui] , where (m-q) + 4k < 1/2(m+4g). i=1

The single ~nbiguous case occurs for

86

g ~ [Ti,U i] respectively. Then we distinguish the one (say) m i=1 which has T 1 in its description.

m = 4g: S1...S

B.

or

F is not orientable. If ~ does not separate and is one-sided, and if by cutting along ~ we obtain a non-orientable surface, then the word is

3.5.7

V1; in this case we must have g > 1.

If an orientable surface results, then

3.5.8

V1...V g (g odd). If ~ is two-sided and the surface remains non-orientable after cutting, then

3.5.9

VIV2, g > 2. If a orientable surface results, then

3.5.10 V1,..Vg (g even).

Now if ~ separates, and the resulting components are both non-orientable, then the word is

k ~ v~, k > 0 ~ t h (m-q) + 2k < 1/2(m+2g). i=1 If one component is orientable, but the other is not, then there are three

3.5.11 Sq+I...S types

3.5.12 (a) So+I...SV$...V2 - 1Vi ....Vg zV ....V g where (g-i) is even and > 0, m - q + 2(i-1) + 1 _< 1/2(m+2g).

If the inequality does not hold for 3.5.12 (a) one replaces Sq+ 1. • .SmV 1 ~ " ..Vi-1 2 by the inverse of the complement in K and reduces cyclically. The word becomes

. . . . 2 -1 Vi+I...VgV i. . .Vg (q > O) (b) Sq1...S11Vg2...Vi+1Vi (c) Vg1Vg21... V -2 i + lVi-1"Vi+l"'VgVi ...Vg_~ ( q = O ) If two curves have different descriptions under the classification (3.5.5-12) then they cannot be mapped on to each other by a surface homeomorphism. Proof. We merely remark that curves which have the same description relative to pos-

sibly different canonical dissections can be carried into each other by a homeomor-

phism, and indeed by the one which carries the canonical dissections into each other. (Curves which have the same description relative to the same dissection may be car-

ried into each other by an isotopy, cf.

87 One can see inmediately that curves with different descriptions 3.5.5-12 cannot be carried into each other by a homeomorphism, because of the characterization of the circumstances 3.5.8, 3.5.9 and 3.5.10 relative to each other and in comparison with

3.5.5~ 3.5.6 and 3.5.7. On the same basis, 3.5.5, 3.5.6 and 3.5.7 are different from

one another. ~ne distinctness of ~ahe descriptions in the cases 3.5.6~ 3.5.11 and

3.5.12 lies in the fact that the description uniquely deter~nines genus and the number

of boundary components.

[]

The descriptions 3.5.5-12 have the fol!owizz properties: 3.5.13 They are cyclically reduced. 3.5.14 They contain no subwords which comprise more than half of the defining rela+1 tion (~(H))- . 3.5.15 I r a

subword occurs which comprises half of a defining relation

l[~(H), e = -+ 1, then for g = 0 this contains the Sm' ~ and for o~ > 0 the T c 1 or V 1 respectively.

3,6

INTERSECTION NUMBERS OF CURVES In this and the next section we consider some homolo~ica] properties of or~en-

table surfaces: the intersection nmmbers of curves and homolosical properties of

mappings between surfaces, in particular., the mapping degree is revisited. Mostly we will restrict ourselves to closed surfaces. 3.6.1 Definition of intersection numbers between curves~ Let F be an orientable surface. We choose an orientation and define: The star a l ' " " a n

at the vertex Q

is called positive if a~-lai+1 (1 _< i _< n-l) is in the positive boundary of a positive face. ],k)wlet ~1 and ~2 be paths and assume that there is a situation -1 ~i = ~ilailai2wi2 (i = 1,2) where the edges aij' i, j E {1,2}, start at one vertex Q.

~.e position gets the n ~ e r

if the pairs (all,a12) and (a21,a22) do not separate themselves in the star

at Q, 1

the edges follow in the order ~11,o21,~12,~22 in the positive star at Q,

88

-1

~11,a22~12~21

is in the positive star.

A similar definition is used if the initial or end points of ~1 or ~2 are i~wolved.

Now assume that ~1 and ~2 are closed curves which have no edge in c o n ~ n . Then

the intersection number V(~l,~ 2) is by definition the s~m of the n ~ e r s

at all

possible positions as considered above.

In a similar way we handle the case where the curves ~1 and ~2 have co~mon

edges. Assume we have a situation ci ~ , ,-1 , , ~i = ~ i 1 ~ i l ~ o ~ i 2 ~ i 2 Where the edges ~ j

Cl = i, ~2 E {1,-1}

and a~j are different (j = 1,2)~ and ~o = ~ i ~

leads from Q1 to Q2" (If ~

consists of one edge only, then ~1 = ~

(T i edges)

= ~1

) ~bw let o o c2 = 1 and consider all'a21 ' ~ ' ~1 ar~ ~2,a12,a22.' ' If bo~h are in the positive stars

at Q1 and Q2' then we adjoin +1 to the siO~ation; if both belong to the negative stars then -1, and 0 if they belong to stars with different orientations. For

e2 = -1 we reverse the signs. Now the intersection number is defined as above as

the sum of the numbers for all such situations. (See the figure.)

I +1

0

7 -1

By easy arguments it can be proved tb~t the intersection number bet~een two

curves ~ and ~

does not change, if ~ is replaced by ~ where ~ is obtained by a

deformation 2.4.2 (a) or (b) over an edge or a face: ~ ( ~ , ~ )

= v(~,~). Hence,

the intersection n,~mber is an invariant of the equivalence class of c~. Now the

following is obvious:

3.6.2 Lemma. (a) V(~l,~ 2) =-v(~2,~1).

(b) .~(~11,e~2) = -,J(~1,~2 ) = w(~1,~21). (c) ~ ( ~ 1 ~ , ~ 2 ) = W(~l,~ 2) + ~(~],~2 ).

89

(d) If ~i and ~ l are homotopic, then w(w~,m 2) : ~ ( ~ , ~ ) .

(e) Let F be the fundamental group of F. Then w induces a skew-symmetric bilinear mapping - again denoted by w

-

~: F x F ÷ Z, (Wl,W 2) ~ V(~l~ 2) if ~i C w.. l

For a fixed y E F we obtain a homomorphism ~': F ÷ Z, x ~ ~(x,y). Since ~ is

abeli~n, ~

vani~es on the commutator subgroup of F, hence ~' i~Juces a homomorphism

of the abelialazed fundamental group, that is of HI(F) , to Z, see 2.7.8. The same armament can be applied to the second variable and we obtain: 3.6.3 Proposition.

(a) The intersection number between c~rves of the oriented sur-

face F induces a bilinear skew-sy~netric form

~: Ill(F) • HI(F) ~ Z. (b) If Sl,.°.,Sm,tl,Ul,...,tg,Ug

is a canonical s~stem of generators of F then

~(si,s k) = ~(si,t j) = w(si.u j) = w(ti,t Z) =w(ui,u l) = O (1_< i, k_£%menta! ~roup of the torus, prove that each

automorphism of Z e Z is induced by a homeomozphism. E 3.20

Prove 3.4.3.

E 3.21

Prove Proposition 3.21.

s s.22

Let T be a torus a ~

tal group. Prove

s~t a c~non;i~cal system of generators for the fundamen-

(a) If a,b 6 Z, (a,b) = 1 then the class of sat b coKtains a simple closed

cur',Te

.

(b) I f a , b , c , d

6 Z and ad-bc = _+ 1 tI~en ~2qere i s a c a n o n i c a l p a i r o f c u r v e s

w i t h t h e homotopy c l a s s e s o f s a t b and sCt d. (e) Converse t o ( b ) . E 3.23

(a) Give examples of simple closed curves y1,y 2 on an oriented closed surface

of genus g ~ 2 that are not homo~opic, but for an arbitrary curve 6 satisfy

,o(¥1~6) = v(¥2,6). (b) Show that Y1 and Y2 are homologous.

I05

E 3.24

Fill in the details to 3.6.1.

E 3.25

Do the ealculatiens ~plicitly to obtain the formula 3.6.5.

E 3.26

Describe homeomorphisms of the surface from 3.6.7 that Lnduce -the matrices

E 3.27

Give the details for 3.G.10.

E 3.28

Prove 3.6.7 (b) also for m > O.

E 3.29

Prove: A continuous mappi~ng f: 82 ~ S 2 of degree 1 or -1 is homotopic to a

(A - D). In case (A) you may restrict yourself to the case where B2,...,B S O 1 0 1 are unit matrices and B 1 is -1 0 or -1 1 , see E 3.19 (a).

homeomorphism, in fact, either to the identity - if the d e ~ e e is 1 - or to the reflection at the equator:

(x,y,z) ~ (x,y,-z),

if the desree is -i.

(Hence S 2 has been identified with {(x~y,z) 61R 31x 2 + y 2 + z 2 = 1}.)

E 3.30

Prove 3.7.3 with tools of algebraic topology.

E 3.31

Proof of 3.7.7.

E 3.32

Proof of 3.7.8.

E 3.33

Give the cohomolosical interprei:at:ion of section 3.7.

E 3.34

Prove that there is no mappin S F

E 3.35

Discuss T: [Fg,F o] ÷ [F, ,Fo] ~

- Fh, S < h, of non-zero desree.

see 3.7.3.

4,

PLANAR

DISCONTINUOUS

GROUPS

in this chapter we consider finitely ge~'erated groups that act on plan~complexes,

construct their canonical fundamental

domains and presentations,

classify than with respect to geometric equivalence that each group with a presentation

and isomorphism.

a~d

Then we show

in one of the canonical forms can be realized

as a group acting on a planar eomplex. The interest in +gnose groups arises from complex ar~alysis where they occur as discontinuous

groups of motions of the non-

euclidean plane, often called P.lchsian groups~ we will deal with this side in chapter 5. Our treatment here is purely combinaroriai

and we prove the group theoretic

propemties of these groups using only combinatorial

4,.]. P L A N A R

arg~mments.

NETS

In this section we classify the plane among surfaces and prove co~i)inatorial versions of the Jordan curve theorem and the Sch~nflies theorem. topological

For the famous

versions of these theorems and some remarks on their history,

[N~wmea 1951]

~d

4.1.1 Definition.

see

chapter 7. Let F be a sKvface. We say that two faces t, ~' are connected by

faces when there is a sequence ~1 = t, t 2, ' ''' ~n = ~' of faces in which ti and ~i+1 have at least one boundary edge in common.

F is called 2-connected if any two

faces of F can be con~nected by faces. A curve ~0 is called separating when there exist faces ~, @' of F which calmot be connected by a sequence of faces where consecutive have a conmon edge not from L0 in the boundary. 4.1.2 L~mma. Let F be a surface and ~ a null-homologous

(see 2.7.1)

simple closed

curve in the interior of F. Then F is separated by ~. Proof.

Let m = Ol...~n, and let w = s I + .., + s n be the associated chain of C 1

(see 2.7). Since m is simple-closed

all the s. are distinct and since w is nulli there is a chain f e. C 2 with ~2 f = w. Let f = Ea.f. where the f. corresI i l pond to the faces #i of F. Naturally only finitely m&ny of the a i are non-zero. Tf homologous

two faces #i' ~j have a conmon edge which does not appear in ~, then since ~f = ~(Ea.f.) = w we havc the equation a i = -ca., where the common edsc appcars in m i ] the boundary of %i with exponent + 1 and in the boundary of Cj with exponent e. If does not separate, one can cormect any two faces without "crossing" m. But that means that all the coefficients

a. have the same absolute value, so that the coefm ficients of the s i in w are divisible by 2. But this contradicts the fact that

w = s I + ... + s n and the s i are distinct.

[]

107

4.1.3 Definition. By a line we mean a connected infinite graph in which two edges emanate from each vertex. A surface equivalent to a complex with one geometric face +1 _+1 $- , one boundary geometric edge ~ and one vertex is called a disk. A planar net is a connected surface ~ with the following properties: (a) 7< is open, i.e. ]E has infinitely ~ y

faces and no boundary.

(b) Each vertex has a fLuite star. (c) Each simple closed path bounds a disk. It follows ~ e d i a t e l y

from (c) that ~1@i;) = 1 ~nd HI~E) -- 0. ~ e

converse also

holds : 4.1.4 Lemra. Let E be a connected open surface with finite stars. If HIGE) = 0 then

is a planar net.

Proof. Let ~ be a simple closed curve, w the corresponding e l ~ e n t are finitely many faces ¢i such that ~(~a i.f.) i : w, with a.i ~ 0

of C 1 . Then there

for the associated

f.. Sinee2~ is Lnfinite and ~} is simple, the a. = + 1. ~ne $i constitute a eomueeted i surface with the one boundary curve ~. One easily reaches the conclusion that the homology of this complex in dimension 1 is trivial. Thus by c o r o l l ~ y

3.2.10 it

constitutes a surface of genus 0 with boundary curve ~}. It is therefore a disk.

4.1.5 L ~ .

[]

A ~ine in a planar net separates it.

Proof. If not, consider two faces which meet along a segment ~ of the line, and take two vertices PI' P2 (possibly after subdivision) which lie on their boundaries but rmt on the line. We cormect thegn by a simple curve ~ which does not meet the line. Further, after subdividing the faces by two edges ~1,~2 where ~i goes from P1 to a point Q on ~, aIld ~2 goes from Q to P2' we add ~1 and ~2 to ~. Then a1~2~ bour~ds a (finite!) disk containing one (infinite!) half of the line.

4.1.6 L ~ a . SinceE' tion, H I ~ ' )

If ~] is a planar net andE'

[]

is related to i~, then 7E' is a planar net.

is related to]E,]E' is open and the stars of]E' are finite. In addi: 0~ so the le~ma follows from le~na 4.1.4.

D

We recall that two nets E and E' ore called isomorphic when there is a one-toone mappin8 f: ]E ÷ IE' which maps faces to faces, edges to edges and vertices to vertices,

in such a way that the boundary relations are preserved, and each part of]E ~

108

has a pre-image. 4.1.7 Theorem. Up to isomorphism,

any ~0o planar nets are related.

Proof. Let ]E be a planar net. By subdividing se~nents ~ I can be converted into a

related net in which no segment is closed. 'Ibis net will also be denoted by E. Let

~o be a face of I with boundary ~Jo . The 2-comnlex K containing all faces, and their bound~ies,

which have ar~ edge or a vertex "n co]m~on with ~o is a finite co~91ex,

since at most finitely many faces meet at each vertex. Cor~sider all the simple closed cLmves on its boundarTj. Each bo'~ds a disk. If we fill in all disks which

contain no face of the complex K in their interior, then only a single simple clo-

sed boundar~y curve ~1 remains, since %o can be connected with each face of the new

complex without crossing a boundary curve of this complex. Let the disk bounded by

~1 be 91 . By repeating this process with 91~ arld continuing~ we obtain a system of disks 9o c 91 c ... of which the i th contains all the preceding in its interior as

well as all edges emanating from 9i_1. Since Z

is connected it is clear that

u }i contains all edges and hence equals ~. Furthermore, we have a system of simple i=1 closed paths ~o,~1,w2,... which do not m~et~ and the ith of which contains all the preceding ones in the interior of its disk. 9i is related to a disk~ thus if one removes the disk 9i fr- O)

£k,mk+ 1 ends there. For

i = 1,...,m k the final point of Yk,i coincides with the initial point of Yk,i+l and

is a rotation cente~ (if m k -> 1). By extending the curves already obtained by cuts • 1~ ~l,...,~g,~g or ~l,...,~g respectively,

beginning and ending at Q, E/G becomes

simply connected. We then have a single face for E/G which we can give the boundary m

- --1

i=1

a.o. z I

g

- --1--1-

q

.

.

.

-.

H ~j Uj K ~kYk,1...Yk,mk+in j=l ~j~j j=l

- --1 gE ~.~. - q~ n. . kYk~l...Yk~mk+lq . ~. ., . Hg ~.a. i:1 I i j:l 3 3 k:l

1

1

or

respectively,

the same way as in 3,2.

All the processes necessary to obtain this canonical normal form may be lifted to and we obtain 4.3.6 Theorem. A planar discontinuous group with compact fundamental domain may be

realized by a pair ~ G )

in which any two faces are equivalent, only the identity

mapping from G leaves a face fixed, and the boundary path of a face has the form: 4.3.7

m

g

-1 _ 1 !

m ff a'a-13_

g

q

ff ~:a -1 ~I T!~. • i=1 z z j=l J J 3

q

k~lq~Yk~l...yk~r~+lnk ] =

-1

ol~ 4,3,8

i=1

z

TI ~.~!

E n~¥k,1.

j=l J ] k=l

-1

"'Yk'mk+lnk

respectively.

Here edges denoted by the same Greek letter and index (e.g. ~ 3. and ~.) are equivalent. i

115

The endpoint of ~ is a rotation center (of order h. ~ 2). If m k > 0 then the final i 1 points of Tk,i (1 ~ i ~ m k) are likewise rotation centers (of orders ~ , i ~ 2). If

m~ = 0 the initial vertex of Yk,1 is not a rotation centre. Any two of these rotation

centers and any two segments not denoted by the same Greek Setter are inequivalent. The path 4.3.7 or 4.3.8 is simple-closed.

7he proof of the latter assertion will occur incidentally in what follows. Exercises : E 4.3,4.

4,4

FUNDAMENTAL DOMAINS

4.4.1 Definition. A connected subcomplex of Z containing exactly one face from each

equivalence class, together with their boundaries~ is called a fundamental domain of aE,0).

4.4.2 Theorem. Each pair (E,@) has a fundamental domain, and fundamental domains are

simply connected.

Proof. a) Let % be a face. By proceeding from 9 to other faces in succession, each

having a boundary edge in common with the complex K already constructed, but inequi-

valent to &ny face of K~ we obtain a connected subcomplex F. Then ~ny face ~' inequi-

valent to the faces of F can only meet faces inequivalent to those of F. Since % and %' may be conmected by faces, such a ~' does not exist. b) Let ~ be a s ~ l e

closed curve in a fundamental domain F which bounds a disk

S not lying wholly in F. Then at least one face t of this disk S does not belong to F. Let ~' be the face in F equivalent to % and x the automorphism which maps %' to %. Now % is in Fx, and Fx is connected. Since F and Fx have at most boundary edges in common, and each edge from ~ : ~S is in F~ Fx, ~nd hence ~x, is contained £n S

and we have Sx c S. This contradicts the fact that S and Sx contain the same, finite, number of faces.

[]

It follows likewise that two images of a fundamental domain have at most a sLmple, non-closed (possibly infinite) path in common. If F is finite, then F is

a disk. In particular, its boundary is s[u~ple and closed. One obtains ~E/G from F by identifying equivalent boundary edges. ~ne boundary path of F uniquely determines

116

E/G (as a s~face). We provide the bo~mudary path with an orientation. 4.4.3 Theorem. At most two directed edges from the same equivalence class appear in

the boundary path of a fund~sental domain. If there are two, then they are equivalent under an orientation reversing automorphism, and no element from the class of their inverse edges appears. If only one edge a appears from an equivalence class, then -1 there is ar~edge equivalent to a in the boundary path, or else a is a fixed edge. [] The proof is Exercise E 4.5. By mear~s of elementary tr~sfo=~mations, carried out sLmult&neously in all Fx

for x e G, we can reach the situation wher~ F is a face with boundary path 4.3. ?

or 4.3.8. Among other things, it follows that this path is simple and closed.

4,5

THE ALGEBRAIC STRUCTURE OF PLANAR DISCONTINUOUS GROUPS The algebraic strmcture of a planar discontinuous group with compact fundamen-

tal domain may be easily derived from the "dual net". 4.5.1 Definition. The surface F ~

the following way :

dual to a surface F without boundary is defined in

(a) Each geometric face ~_+1 of F corresponds to a vertex ~

of

F ~.

(b) Each geometric edge 0 -+1 of F corresponds to a geometric edge a*+-1 of F~. (c) Each vertex P of F corresponds to a geometric face p,_+l of F~. (d) ~ne boundary relations are carried over. ~nis means in particular: if P is a boundary vertex of a, then a* is a boundary edge of the face

P* (if an,...,a ,aA is the star of P in F, then oi...a* is a positive boundary ± n i+ 1 path of the pair P*- , the oppositely oriented star of P corresponds to a positive boundary path of the oppositely oriented face). If a is in the positive

boundary path of ~, then ~* is the initial point of a* and the final point of (o-1)* = a*-1 . If P is in the boundary of ~, then ~

4.5.2 Lenma. The dual complex

is in the boundary of P*.

F~ of a surface F without boundary can be identified

with a complex of the same surface.

117

P~oo] ~ (see [Reidemeister 1932, pp. 133-136] ) W e

first construct a subdivision of

F ; this is described intuitively rather than formally. Each edge a is divided by a new vertex a ° into t~4o edges a' ~a". ~nen in each face ¢ we take a vertex ¢* and

connect ~* with all the new vertices a ° which belong to edges in the boundary of ¢.

The complex F' obtained is equivalent to F. Let P be a vertex of F. 7hen there are as many faces containing P in their bounda~

as the star of P in F contains edges. We join all the faces of F' which have

P in its bo~uda_~y into one face P*° Here we loose all edges of }~ with P in its bousiary, as well as P itself.

A vertex of the fomn a ° meets exactly two edges, say from o . o in their boundary. We drop a and combine

and ¢3 in the faces 9i~¢ j which have a

¢i these two edges into an edge a*. The construcjtion is clear from the figure.

I I

... "~

I I

~

I

p. ~ ~ ~ ~ .~-

I

..

~

I

I I

~--- ... -.. ~

I

...

I I

I I

I ~

~

We now assume that the g~oup G acts o n e particular,

I

I I

[]

simply transitively on the faces, in

that G has a compact fundamental domain. The group g acts o n E * ,

but

now simply tr,ansitively on vertices and perhaps rotationally on faces. At each ver-

tex we label the edges originating the_re with symi~is +1 +i +I , i E k , Ck, j according to the following rule, where the upper line contains the symbol from 4.3.7 or 4.3.8 and the lower line contains the symbol for

the corresponding dual edge: -1 -1 -1 -1 a'm a.m ~j.~j ~j ~j' ~ j ~'3 n'k Yk,j nk

SiSil

Tj U7 3 i T3[ i U.3 V.3 VT 3 i Ek Ck,j Ek 1. The star of edges emanating from a vertex then reads:

4.5.4

1,s 1 .....

. . . .

g

g

g

g

EI'CI,I'" " ' 'C1,m1+1'E11'" " "'Eq'Cq,l'" ""Cq,mm+ 1,Eq I or

118

....

....

..... C

respectively. In what follows we use X to denote a "general" symbol. A pair of oppositely directed edges f r o m E ~ receives either inverse s y ~ i s

or

else the single symbol Ck~ j , because G acts s~mply transitively on the vertices of

and has no fixed edges other than the ~k, i" ~nerefore, only edses denoted by the

same symbol (includin~ e ~ n e n t )

are equivalent.

We now distinguish one vertex of ~ ~nd call it 1. Every o~her vertex of E ~ is

associated with the unique automorphism of G which carries it into 1. When the edge labelled X emanat~n~ from 1 leads to the vertex x, then we let X correspond to the -1 g. lh~e edge labelled X -1 emanating from 1 then leads from 1 to x

elmnent x of

and thus X -1 corresponds to the inverse element. Equivalent edges o f ~

obtain the

same symbol. Thus each path i n E ~ corresponds to a word in the symbols X, and con-

versely, each such word W(X) uniquely determines a path when the initial vertex is

given. We say that this path is obtained by tracing W(X) from this vertex. If we now always write X for the corresponding element x we obtain a product W(x) of elements

of G~ hence an element of G. The final point of the path beginning at 1 for the word W(X) is the vertex W(x) ; namely, if W(X) = X1...X n, then x I 6 G maps the final ver-

tex of the edge denoted by X 1, which begins at 1, to 1. As a result, the path deno-

ted by X2...X n beginning at x I is mappedto the path denoted by X2.. ,Xn beginning at

1. It follows that the el~z~ents x corresponding to the X are generators of G and the

relations correspond to the closed paths em,anatin~ from 1. Since only edges in]E~ denoted by the same symbol X are equivalent under @,

when the word W(X) is traced from a vertex we either obtain a closed path on all

occasions, or never. This now facilitates the determination of defining relations.

Namely, a closed path may be decomposed into simple closed subpaths, possibly with

approach and return paths, and spurs. But a simple closed path in a planar net is a

product of paths consisting of an approach path and the bo'~ndaz~y of a face. Spurs

each contain a subpath which runs out and back across a single edge. Therefore we

obtain the defining relations for

G by traversing a path out and back across an edge

or by traversing the boundary paths of the faces of ~

and replacing the symbols X

in the resulting word W(X) by the generators x or a path out and back across a seg-

ment. Of course the path out ar.d back across a segment only ma~es a contribution

when the two directions are not denoted by inverse symbols, i.e. if it is on an axis of reflection. Since G acts simply transitively on vertices we need only consider

119

faces and edges which have 1 i~- their boundary. Even now: some W(X) cain appear se-

veral times (up to cyclic L~terchange). We do not attempt to abstractly characte~oize a minimal system of defining relations, but dete~Inine only the words W(X) which

appear.

We call the star 4.5.4 or 4.5.5 positive when its center vertex corresponds to a~n oriertation-preservin4~

autemorphimn. At all other vemtices the oppositely

oriented star will be reckoned positive. We then obtain the boundary paths of faces by following each successive ec~e with the neighbour of its inverse, taking the lyositive sense in the star of its final point. In addition, of course, we must consi-

der the orders of rotations, as a result of which the boundary path of a sur-

face piece h ~ the form (R~(s))k.

h.

From 4.5.4 and 4.5.5 we obtain the boundary paths

i ' (Ck,j+lCk,j)

1 < j 3. As before

we take a p and a k such that ( p ~ 2 h l . . . h )

= 1 and (p~< - 1) - 0 nod 2hl...h

m

m

. 1%~en

in SLF( 2 ,pk) we take elements s3,s4,...,sm,ul,tl,. ' ~' ' ' ' .. ,u'g, t'g such that s l! has order g ' . .s'm i=1 K [tl,u £] is unequal to 1. As in the foregoing proof we find h i and w = s 3.

.....2 i n ~ F ( 2 , n ~ k) where s~' ,h 2 respectively and s~ s~ has the Sl,S ± and s~ have orders h~.... -i ,, ,, -i . . . . smm~e trace as s . d e f ~ e additional free generators

which only occur, in the "long" relation. The subsurfaces Si~ 1 ~ i ~ n, with their markings give further generators (if there are dark curves) ~ d

relations. A full

system of generators smd defining relations is given in 4.11.3,4. Here the genera-

tors 4.11.3 (c,f) and defining relations 4.11.4 (d,f) belong to the "holes". 4.11.3 Generators.

(~)

S, 1

(b,+) tk,uk

k = 1,...,g,

(b,-)

vk

k = 1,...,g,

(c)

e. D

j : 1,...,r, (here r 1 > 0 if the fundamental doKmin is not compact)

(d)

e.

j = r1+1~...,r1+r2~

(e)

e.

j = r1+r2+1~...,r1+r2+r3;

(f)

c. ]Pq

]P

j = rl+r2+r3+l~...~r;

p = l~...,mj+l~

p = 1,...,kj; q = 1,...,mjp.

139

4.11.4 Defining Relations. h-

(a)

s. l i

(b)

2 c. : 1

(e) (d) (e)

(f)

:

I

j : l~...~m% h i > 2, j : rl+l,...,rl+r2,

2 c. = 1 3P 2 c. : 1 3Pq

j : r1+r2+1,...,r1+r2+r3 j : rl+r2+r3+l,...,r;

h. Jp = 1

(C,1pC_p+ )]]

; p = 1,...,mj+l,

p = 1,...,kj; q : l~...,mjp:

j = r1+r2+1,...,r1+r2+r3

~ p = 1,...,mj{

h. > 2, m. > 1, IP J

h.

(ejpqCjpq+ 1 ) 3Pq

1

j : rl+r2+r3+l,...,r~

p = 1,...~kj~

q : 1,...,mjp-1~ kj _> 1, mjp -> 1, -1

(g)

c.e.c.e.

(h)

c ' l e ' c " m +1 eT1 = 1 3J, j

(i,+) (i,-)

JJ

33

=1

j : r1+1,...,r1+r2,

m r g H s. ~ e. i=l i j=l 3 k=l m

r

g

j : r1+r2+1,...,r1+r2+r3~

[tk~Uk:l

~ s. ~ e. H v i=1 l j=l 3 k=l

= i~

=1.

Here (b,+) and (i,+) or (b,-) and (i,-) are valid together. The + denotes the case

where the surface E/0 is orientable, in the "-" -case the surface is non-orientable. 4.11.5 Theorem. 0 has the canonical presentation

.

The proof of this theorGm is left to the reader. _The algebraic s ~ u c t u r e of

the groups has been described in [Hacbeath-Hoare 1976] ~nd [Ziesch~ng 1976].

From

the geometrical interp:~etation of the generators a~nd the n~mbers re,g,.., involved

it is clear that presentations with essentially different numbers describe inequi-

valent groups of actions. Of course~ one may permute the orders h. ~ and the systems I

(hjl ,...,hjm j) or (hill, hj12,...,hjkjmjkj). A~other question is the classification with respect to (algebraic) isomorphism. From the presentation 4.11.5 it becomes

clear that the group is the free product of cyclic groups and groups which are ge-

nerated by reflections connected by rotations, (the joining long relat:ions 4.11.4

(i) can be omitted). Hence, the classification can be done by going to a free pro-

duct decomposition with indecomposable factors, which are uniquely determined up

to isomorphisms and permutation, see 2.3.14.

Exercise: E 4.16

140

4,12

ON D E C O M P O S I T I O N S

OF D I S C O N T I N U O U S

GROUPS OF THE PLANE

The decomposition of a discontinuous group of the plane into free products

mentioned at the end of the precedi~ig section does not cor~,espond to a 'geometric

decomposition of the action'. But the decompositions into f 2. T h e ~ l b e d d i n g s A ~-~0. c a n be r e a l i z e d b y m a p p i n g s g~: S + X i , a n d we 1

obtain the mapping cylinders X_l u 81 × [-~,0] and S 1 × [0,i] U X I . We glue those together

a t S 1 = S1 x 0 a n d o b t a i n

the

s l u i c e X = X_I u S 1 × [ - 1 , 1 ~

u X1, s e e f i g ~ p e :

From the Seifert-van Kampen theorem it follows that ~l(X) = 0 and that the fm~damental groups of X_I,X 1 and S 1 ape embedded into 0 the same way as 0_1, 01 and A. The space X is a K(0,1)-space, induces an isonmrphi~between

hence we may define a mapping f: M ÷ X which

the fumdamental groups ~I(M) and ~I(X). Since Ln a

free product with amalgamation an element of finite order is conjugate to an element of a factor we may assmne that the botmdary of any disk D. is mapped into X_I or X1, hence it does not intersect $1.

3

Next we deform f to bring f-l(S1) into general position~ we denote the resul-

ting mapping also by f. Then f-l(s1) consists of a numi)er of s ~ p l e

closed curves

in M,none of which intersects the hounda~7~ of a disk D. or is nullhomotopic. A ] slight generalization of the Baer theorem 5.11.1 is that tv~o disjoint sLmple

closed curves on M botmd an armulus if they ape homotopic~ but not null-homotopic

and do not cross the ~D.. By a deformation of f we can get rid of a pair of neigh-

bouring curves from f-l~s1). Finally we end with a mappir~, again denoted by f,

wh£ch has at most one curve in f-l(s1). If f-l(s1) is eJnpty then H would be mapped into one 'side' of X, thus ~l(H) into one of the factors ~l(Xi ) = G i. This contra-

dicts the assu~.ption that f induces an isomorphism of the fuundmmental groups, since each factor ~I(X.) has infinite index in 'r~l(X). Next we use the fact that a simple 1

142

closed curve in M that avoids the 8D.l is not homotopic to a proper power of another curv%see exercise E 3.10. Therefore f maps f-l(s1) homeomorphically onto S 1. ~_rrom

the

Seifert-van Kampen theorem it follows that f-l(s1) separates M into two parts

M_I and M 1 which are mapped to X_I and X1, resp.~ and f defines isomorphis~ between the fundamental groups of M. and X.. i I

Now we have proved (b). The assertion (a) is a simple consequence of (b) and

the classification theorem 4.6.3 for planar discontinuous groups.

[]

The argument becomes more interesting and complicated when the quotient sur-

face is non-orientable because of the appearance of Mbbius strips; this is the

ma~

part of the proof in [Hendriks-Shastri

1978].

The algebraic decompositions of planar discontinuous groups into free products with

amalgamation on one hand and the geometric decomposition of the action on the other,

were considered in [Zieschas~g 1976] and [Lyndon 1978]. The geometric problem is

solved, but for the algebraic question there are some u:nsolved cases. For ex~nple,

the groups , k _> 1, have indecomposable actions, but they can be decomposed into proper free products with amalgamated subgroups, see exercise

E 4.22.

4,13

PLANAR GROUP PRESENTATIONS AND DIAGRAMS

In combinatorial group theory it is sometimes convenient to represent a group

with given generators by a 1-complex, the Cayley diagram ar Dehn Gruppenbild. In

fact we have already used these diagrmns, without saying so~ in the solution of the

word problem in 4.9. The groups we a~e interested Ln have nice presentations which correspond to useful 2-complexes. We have used these complexes in the proof of the existence theorem 4.7.1.

The treatment would be much sLmpler if we were to consider only planar discon-

tinuous groups without reflections. This case ~ms considered in [Zieschang 19 6 6],

following an idea of K. Reide~neister, see also [Reidemeister 1932]. The more general situations have been dealt with in [Lyndon-Schupp 1977]

and

~rm~berg-Ziesch~ng 1979]. We repeat the approach of the latter here~ applying the refined Reidemeister,-Schreier process of 2.2, and continuing to use the notation

of that section.

143

4.13.1 Definition. Let G : . 9~m letters v, w E S U S -1 are called algebraic -1 neighbours iff v precedes w in a word R e where R E R and s E {1,-1}. Here R e is considered as a 'cyclic' word. . For instance~ if R = s k, k _> 1, then s -1 is an algebraic neighbour of s. If v

is an algebraic neighbour of w then w is an algebraic neighbour of v, too

. In the

study of crystallographic groups of the plane it is convenient to characterize the

reflections already in the presentations as follows, cf. [Hoare- Karrass - Solitar 1973], [Lyndon-

Schupp 1977, III. 7, 8].

Let G = ):= 2"(IS'E

+ IS"I - ~

RcR'uR"

1

- 1).

Here we use the order R" < R' for R ~ E R', R" E R 'T. (For the definition of Stab cf. 2.2.4.) In the definition of the sequence of neighbours we avoid sequences with periods and assume that each letter occurs only once in the set of sequences, Planar, presentations appear in the study of discontinuous groups of motions of the euclidean or non-euclidean plane and these are the main examples: 4.13.7 Tr~eore~m. If G is a discontinuous grou~ of the plane then any fundamental do-

main defines pairs of generators (the transformations that move the fundamental domain to a neighbour with a common side) and defining relations (which correspond to the inequivalent vertices of the fundamental domain). If the generators that correspond to reflections are put into S", and from each other pair one member into S ~, the result is a planar presentation.

145

This can easily be checked on the canonical presentations of finitely presented discontinuous groups, see [Macbeath- Hoare 1976], [Ziesch~ng 1976] { the generali-

zation to other 'geometrical presentations' cm~ be proved using 4.14. The fundamen-

tal domain is conpact iff there is only one sequence where the first a:nd the last

letter are neigbbo,ars. Next we will show that each pl~nar presentation can be obtai-

ned that way, except when the Kroup G is finite. (Then the preseatation belongs to

a discontinuous group of the sphere. ) For the proof we use modified Cayley diagrams, see [Lyndon -

Schupp 1977, p. ]34]~ and we will first repeat their construction:

4.13.8 Construction of modified Cayley diasrmms. Let G be presented as in 4.13.2.

Now we construct a 2-complex as follows:

(a) Let C ° consist of vertices which are in a 1-1 correspondence with the elements of

G

and we assume that the vertices are labeled by the group elmn~nts.

(b) To each pair (g,z)~ where g 6 G and z e S' U S '-1 O S" corresponds exactly one

directed edge Which starts at the vertex g and ends at the vertex gz. The inverse s -1 s -E I' edge (g,x i) equals (gxi,x i ) where i E ~- 6 {1,-1} and the inverse edge (g,yi)

--1

equals (gyi,Yi) for i 6 I".

C 1 consists of C° and the edges described above. Note that (g,yi) and

(gyi,Yi) = (g,yi)-1 are equal as undirected edges. (With this exception C 1 is the

usual group diagram correspondiP~ to the generators S' U S". ) We label the (directed) edge (g,z) with the symbol z. To each pair (g,w) where g 6 G and w is a word in the

generators now corresponds a u~iquely determined path ~ in C 1 with initial point g such that the edgepath is labeled by w { the final point is the vertex gw. We call the realization of w at g.

(c) To each pair (g, R~) where g 6 G~ ~ 6 {1~-I_} and j E J corresponds an oriented

face D(g,R~). The boundary J of D(g,R~) is the realization of R E.] at g and the face

with the inverse orientation equals D(g,R[~). Two faces D(g~R~) and D(g' ,R~) which have the same bomndary are considered to be J identical. (Here Jj 6 J is fixed, but

the direction and the initial point may be altered. ) The complex C(S, S'; R) con-

sisting of C 1 and the faces described is called a modified Cayley (or group) diagram. 4.13.9 Remark. (a) The group G acts on the 2-conplex C(S', S"~ R) as a group of

autommrphisms in the usual way: for a fixed g E G the transformation tg correspmnding to

g

maps the vertex g' to g'g, the edge (g' ,z) to (g'g,z) and the face

D(g' ,R~) to D(g'g,R~). The boundary of a face ~s mapped to the boundary of the image face etc.

(b) ~IC(S' ,S" ;R ') : 1.

4.13.10 Definition. The modified group diagram C(S' ,S"~R') is called planar if the complex can be embedded into the plane or 2-sphere (if G is finite) in such a way

that the group {tglg E G} is induced by a discontinuous group of homeomorphisms of

146

the plane or sphere. If IS' o S"l :

=

we embed only the 'complex' C(S', S"; R')

without the vertices. (These can be considered as 'ends' on the 'bo~±ndary' of the rmn-euelidea~n plane. ) Our next aim is the following theorem: 4.13.11 Theorem. if be a pla~nar presentation of the group G that determines a closed sequence of neig?fyoured elements, see 4.13.6. Let R C R' and E

x e S' be

such that R = vx w~ where v,w are words in the generators different

from x and e e {1,-1} . Now let ~

= S~\{x} and replace x in all relations of R~\{R}

by (wv)-e; let kn denote the system of relations obtained for the generators ~' U S". The replacement of the presentation

be connected as in 4.14.14. If one of the presentations closed sequence of neighboua~s

is planar and determines a

then the other does also.

[]

4.14.16 Len~ma. Assume that the presentations

0~ on the euclidean plane if ~(O) : 0 and on the spher~ if ~(O) < 0.) The number ~(G) can be determined purely algebraically: A group that contains

orientation reversing elements is not isomorphic to a group consistir@ only of orientation preserving mappings: hence~ the s~'~group of orientation preserving elements is characteristic, see Corollary 4.8.5. Now the numbers m~ hi, h/k , m1 and q can

be determined and finally g and the type a.

4.14.22 Theorem of the Riemamm-Hurwitz-fo:eT~ula. Let G be a discontinuous group of motions of the non-euclidean or euclidean plane with compact fundamental domain and H a subgroup of G of finite index. Then I~(H) = [ G : H ] ' # ( G ) .

The proof of the theorem is a consequence of the following proposition and the

fact that each discontinuous group with compact fundamental domain contains a sub-

group of finite index which is isomorphic to the fund~nental group of a closed or,ier~ table 2-manifold, see Theorem q.10.1 (The last statement can also be proved purely algebraically.) 4.14.23 FWoposition. Let G be a discontinuous groz~ of motions of the non-euclidean or euclidean plane and let U be a subgroup of finite index that is

isomorphic to

the fundamental group of a closed orientable surface of genus y. Then 4X - 4

: p(U)

:

[G:U].~J(G).

Proof. We apply theorem 4.14.1 and determine the numbers of generators and relations in the presentation 4.14.5. As U is torsionfree, S"(U,K) : @ : R"(U,K) for any system of coset representatives. Now, by 2.2. 4.14.24

IS'(U,K)I

: [G:U]'(IS'I : [G:U].(m

+ IS"I - 1) + 1 g

+ ag

+

Z

/:1

(m2+1)

+ q - 1)

+ 1,

156

where we used the presentation of G and put into S", the other geaerators into S'

the elmr~ents cl~ k

The partial order in the set of relations is given in 4.14.2. Now we have to

dete~nine how many relations in R'(U,K) are obtained f~nm a given relation R 6 R' U R". We will prove:

4.14.25

relation

e£k 2

s ihi

X(R,K)

2

hi

(c1 kC/, k+l )

1 c£1~ c £~m£+i~e-1

i

2

2hlk

where Z denotes the long product relation 4.14.20 (d,~). For the proof we will apply l e m ~ 2.2.5 and we have to determine the orders of

the stabilizers. By simple calculations it follows that

e(Stab 2 ) = { l ~ C l k } ' e(Stab h.) : {saNO L a < h.}~ iL 1 CZk siz ab (ClkC/,k+ 1 ) ClklO -< a
214, then N(p+l) = 8(p+4), ([Accola 1968]). Other cases in [l(iley 1970].

(d) NA(Y) = 4(y + 1). This result can be extracted from [M~aclachlan 1965]

according to [Maclachlea~ 1969, proof of Theorem 3]. See also [Accola 1968 (5)].

(e) The maximal order of a cyclic group acting on the closed orientable surface of genus ~ ist 2(2X + 1) [Wiman 1895/96], [Har~zey 1966]. From the second paper

we take the example that shows that Z2(2y+1 ) acts on the surface of genus T. There

is an epimorphism

160

2 2x+1 2(27+1) D° : < S l , S 2 t S 3 1 S l , s 2 ~ s3 ~ s]s2s3>

* Z2(27+1 )

such that the kernel is isomorphic to the fundamental group of a clo~ed orientable s~.~face of genus y. (Proof as Exercise E 4.23.) Automorphisms of Riema~! surfaces are the subject of many papers. We mention here some more, especially those which contain results that can be proved without usiang ri~e complex an~_lytic structua~e~ but o~&y combinstoria! a r ~ m e n t s based on the Riemann-Hurwitz formula 4.14.22: [Accola 1970, 1971], [Greenberg 19BO, 1968], [Jones - Sinserman 1978], [Haebeath 1961' , 1965' ] ~ [~aclachlan 1971], [~atanson 1978] [Sah 1969], [Sinserman 1974'~ 1976], [T~sr~!mni~ 1978]. See also the paper mentioned in 4.15. Exercises: E 4.18-23.

4.15

FINITE GROUPS ACTING ON CLOSED SURFACES As we have seen in 4.10, each pl~nar discontinuous group G contains a normal

subgroup F of finite index that is isomorphic to the fundamental o~roup of an orientable surface F. If G has compact fulJamental domain then the s ~ f a c e

F is closed.

We carl interpret F as factor space Z / F and the factor' group A:= G/F acts on F. Conversely, if A is a finite group acting on the closed orientable surface F then, by lifting the action to the universal cove~, we obtain a plar~ar discontinuous group G; here we assume that F is not a sphere. In 4.14.24-26 we have given relations between the genods of F ~nd the order of A. Let us now consider the effect to the action to the groups associated with F. Since the basepoint is not fixed by the action, the effect on the fundamenta£ group is not properly defined. This is better with the homology group, which is the abelianized fundamental group. Our basic result ist the following: 4.1.5.2 Proposition. Let @ be a planar discontinuous group with compact fundamental

domain and let F ~ O be a normal subgroup which is isomorphic to the fundamental group of a closed orientable surface S

of genus y > 1. Then [G:F] < ~ and the

action of G/F on F/[F,F]:

y'[F,F] ~ x-lyx-[F,F],

x C O, y C F,

161

is effective, i.e. x-lyx[F,F] : y-IF,F]

for all y •

F

x 6

~

F.

L e t us first prove that [G:F] is finite: 4.15.2 Lenma. Let G be a p l a n ~

discontinuous group and N ¢ 1 a finitely generated

normal subgroup of G. Then [G:N] < ~.

Proof. Because of 4.10.8 we may restrict ourselves to the case where G is the fundamental group of a closed orientable surface. If [G:~J] = ~ then N acts o n e

with

non-compact fundamental domain and is a free ~roup as follows from the presentation 4.11.3, since m = O and r = r I > O. Dat the fundamental ~ o u p

of a closed s~m~face,

different from S 2, is not free, see E 2.18.

[]

Proof of 4.15.1. Assume that the action of G/F on F/[F,F] is not effectffve. Then there is a x 6 G, x ~ F such that x-lyx C y.[F,F] for' all y 6 F. We may assume that the relative order of x is a prime p: x p C F. Let H be the subgroup of G generated by x arid F and let ~: H ~ Z Kern ~ = F and the sequence I+F

P

be the ho~)m~rphism that maps F to O ~nd x to

~ H + Z

P

1.

Then

+1

is exact. Since [F,F] c Kern ~, the sequence 1 ~ F/[F,F]

+ H/[F,F]

+ Z

P

+

1

is also exakt. Since x operates trivially on F it follows (E 4.23) that (1)

H/IF,F] : Z 2 7 ~

B

where B :

i 1

P

or

Hence, [H,H] c [F,F], thus IF,F]

:

[H,H].

The group H is a planar discontinuous group with compact fundaJnental d o ~ i n .

There-

fore it has a presentation as in 4.5.6. First, let us assum~e that x presem~s orientation. Then H ~has a presentation with q = O and h i = p (1 _< i < m). By abelianizing H we deduce from and (1) that g = y. From the Riemann-Hurwitz for[~ula 4.12.23 and the definition 4.12.21 we obtain

162 4y - 4 = p- [2m(1 - ~) p + 4~o - 4] 2 - 2y = m, since p # 1. This contradicts

the assumptions

y ~ 2 and m ~ O.

Now let x be o r i e n t a t i o n reversing.

elements.

T h e n p = 2. If H has the p r e s e n t a t i o n

then q 2 1, since the v. anJ c. By abelianizing

. are the only

it follows that

q = ~, m = 0 and v : g.

If H has a p r e s e n t a t i o n

then there ~ e

the

following two possibilities:

(3)

q : 1, m : O~ 27 = g,

(4)

q = O, m : 1~ 27 : g - 1.

For the cases

(3) and (4) we obtain from the Riemamn-Hur,witz

mI (1-2) 4~ - 4 = 2.[E k=ff hlk

+ 2 + 2aS - 4] where a =

In]x)th cases, a g : iT, hence the above equation mI k=l

f 2 L 1

formula 4.12.23 that

f o r t h e c a s e (2)

for case

is equivalent

(3).

to

(1 - - ~ 1 ) + 27 : O. hlk

This e q u a t i o n contradicts

hlk _> 2~ 7 > 2. In case (4) we have

4~ - 4 : 2 " [ 2 ( I

- __i) hl + 2g

4]~

2 w h i c h is equivalent to 0 = 27 + 2 - ]~1' a contradiction.

Remark:

T h e case w h e r e x reverses o r i e n t a t i o n

o r i e n t a t i o n reversing are h o m o l o g i c a l

4.15.3 Corollary

homology g r o ~

c a n easily be excluded,

changes the sign of intersection

since a n rulers

which

invariants.

By p r o j e c t i n g

y ~ 2 and let

transfomnation

[]

to the factor surface F : E / F w e obtain:

([Hurmitz

1893]). Let F be a closed orientable surface of genus

A be a finite group acting on F. Then the induced action of A on the

Itl(F)

is effective.

[]

163

See exercise E 4.25. Next we deal with another approach to finite ~roups of mappings on a s. Then the projection F + F

O

O

denote the

is a branched coveri~z, see 3.3.1.

Let m denote the number of branch points in F° and g the genus of F° . If we lift the action of to the universal cover ~ of F we obtain a planar discontinuous group O which has a presentation as in 4.5.6. But now h I : ... = h

: p a~1 q = O. Moreover, we have an e~morphism @: G -

Zp with

Kern e = ~I(F), hence, e(s i) = k.~ wiJ~ (p,k i) : 1. Next we normalize the generators of G with respect to e an~d obtain: 4.15.5 Le~n~a. G has a presentation P G : O~ ~(F) is generated by elements

xl(t j), x£(~j)

1 S j ~ g~ 0 S ~

×l(s.) l

3 S i ~ m, 0 N £ 4 p-2

4

p-1

such that ~t(F) has a single de~%ning relation in which each generator and its inverse occurs exactly once. Hence,

2y : p.2g + (m-2)(p-1). Further, for each i 6 {3~...,m} we have

p-1 H

£:0 or

×

lk.

~(s.) : 1,

±

in homology:

s i + ~ ( s i) + ... + ~$P-I(si) - O. (b) If m : O, ~I(F) is generated by

×£(tj), ×£(uj)

2 s j _< g, 0 6, and p = n' = 2.

%]]us only the case where p .: 2 and n is a power of 2 remains:

Let n = 2~. As before, g = O, herme:

2 2 G :

and ~I(F) = Kern ~ where ~: G + Z2~ s i ~ 1. Using the Reidemeister-Schreier 2.2.1 we obtaLn

~I(F) :

method

with x i : sis~ 1' Yi : SlSi

where each generator and it9 inverse occurs in R exactly once. Hence they form a basis for HI(F). The action of G/~I(F) on ~I(F) is given by -1 -1 Xi ~slxisl = Yi' Yi ~ slYiSl = xi, hence, with respect to the above basis the action of ~

in HI(F) gives a non-trivial permutation of the basis vectors. ~ u s

the action does not

induce

the identity on HI(F, Zn).

D

4.15.$6 Corollary. Let F be a closed orientable surface of genus y ~ 2 and @: F °+ F a homeomorphism of finite order. (a) Let the order of ~ divide

n and let p be the smallest prime dividing n. Assume

that some homology basis of HI(F) contains more than 2y-1 + 2 elements c with P ~ ( c ) = c. Then ~ is the identity. (b) If there is a basis of HI(F) that contains more than y + 1 elements fixed by ~,

then ~ is the identity.

Proof as exercise E 4.27. Finite groups of symmetries of surfaces have mainl]y been studied as groups

of cor~ormal self mappings of Riemann surfaces.

So Theorem 4.15.3 is from [Hur-

witz 1893]. The complex analytic concept is also basic for [Accola 1967] and

[Gilman 1977]. In this chapter we have developed a combinatorial theo~y of planar

discontinuous Stoups, including the Rie~rm-Hurwitz

formula, hence, we could

translate [Gilz~n 1977] into the language of the combiI~atorial to[~losy a~nd prove the results in 4.15.4-16 without using complex analytic theorems.

170

The proof of proposition 4.15.1 is obtained in collaboration with M. Hellis

and U. Schadowski (Bochum) in the study of (euclidean) crystallosrapic groups,

see [Mellis, Schadowski, Zieschang, preprint]. For more literature on the theory

of automorphisms of Riemanm surfaces see the end of section 4.14 and chap. 6. In addition:

[Gilman 1976, preprint], [Macbeath 1973]~

[Moore 1970~ 1972].

Exercises : E 4.24-27.

4,i6

ON

THE

RANK

OF

PLANAR

DISCONTINUOUS

GROUPS

The (free) rank of a group is the minknal nunJ:~er of elements necessary to se-

nerate the group (2.1.11). 1~me @rushko theorem 2.9.1 states that the rar~ behaves

additively with free products. This is obviously not the case with free products

with amalgamation. J. Nielsen posed the problem: dec:ide whether the planar disconti2 3 5 7 11 nous group in fact needs at least 4 generators. In a more general version, we may ask for the raft< of an arbitrary

planar discontinuous group. Using the Grushko theorem this problem can be reduced

to simpler ones if the group does not have a c o m ~ c t fundamental region. The problem is also solved for groups with cornnact fund~nental region, if they do not contain reflections, by the following theorem:

4.16.1 l%~eorem. The group

G =

for the Reidemeister-Schreier method. To C we

app]y 4.16.1 with 2m and g-1 instead of m and g. For g > 1 the raRk is

2(g-1) + 2m - 1; for g = 1

C has an even nine G = 0 or g :0, m ~ 3. h~ h m The group @ : 3, n >_ m, a.,b. k 2~ G : , b i i b l m ± m i m

H = and f: G - H an ep~mnorphism. TY~en: ~i -i (a) f(s i) : v i Xk. v I for 1 _< i < m, i (b) m = n,(~i,bk.) = 1, and i

(c) if n>_ 4, then ~kl "'' m p is a pelmutation. 1 km (For the proof use the rank theorem 4.16.1.)

177

E 4.33

Let G and H be as in E 4.33, and let f: G ~ H be the epimorphism with f(s i) : v i Xk~ v i . ]if Kern f is torsior. Prove: b (SlS2)a> and If = 2. E 4.37

(a) Let F be the f~ndamental group of a closed surface F. Suppose that F contains a finitely generated non-trivial normal subgroup N. Then [F:N] < ~ or F is a torus or a Klein bottle. (b) Discuss the same question for arbitpary su~faces~ see E 2.9, and for normal subgroups of planar discontinuous groups.

(1%~e exercises E 4.30-36 are from [Zieschang 1976'] ; E 4.35 can also be obtained from [Knapp 1968] where further examples of this type can be found.)

5.

AUTOMORPHISMS OF PLANAR GROUPS

In this chapter we prove that each automorphism of the fundamental group of a surface is ii~uced by a hoii~oH~xphism (Dehm-Nielsen theor~r~) and we characterize those homeomorphiavns which induce the identical automorphism in the fundamental group (Baer theorem). In this chapter we use mainly combinatorial group theoretic arguments, in contrast to the proof of the Dehn-Nielsen theorem in 3.3. The advantage of our approach is that we can also prove a sgnilar result for non-orientable surfaces and planar discontinuous SrnUpS

5,1 PRELIMINARY CONSIDERATIONS If E is a planar net, @ a group of automorphisms on ~ and ~ an automorphism of 0, then we say that "a can be realized by a homeomo~phism" when ~here ape two sub-1 divisions I' and E " of ~ and an isomorphism n: ~' ÷ E " with a(x) = n xn, x E @. The operation of @ o n Z

extends in a natural way to Z' and E".

We call n a homeomorphism and carry over this concept to surfaces. If one takes~ say, the piecewise linear theory as basis, then one can obviously choose n to be a piecewise linear homeomorphism. We shall prove that each automorphism of a pl&nar discontinuous group without reflections can be realized by a homeomorphism ( ~ e o r e m

5.8.2). If @ is a surface

group (i.e. there are no generators s.) then a subdivision ofiE induces one of l E / O and n goes over to a homeomorphism of E/G. Each automorphism of the fuundamental group of a surface F can therefore be realized by a homeomorphism of F. This result is what we have called the Dehn-Nielsen theorem 3.3.11 [Nielsen 1927]. Of course, when we speak of automorphisms of the fundamental group we must keep in mind the basepoint of the group~ we must therefore insist that 4 leaves the initial point of G fixed in I~/0. If we also bear in mind the rotation centers onE/@,

then the general theorem may also be expressed as a theorem about surfaces.

Namely, a homeomorphism of lE/0 which permutes the rotation centers and leaves the basepoint fi~ed induces an automorphism of O. If we remove a "small" disk from E / 0 around each rotation center and around the basepoint t3~en the fund&mental gro~an of the perforated surface is the free group G in the generators $1:$2,...,S m and T1,U1,...,Tg,U ~ (resp. V 1 .....Vg) "of @". An automorphism a: 0 ÷ 0 which permutes the path classes of boundary curves corresponding to rotations of the same order (or maps them into their inverses) and carries the cuts of a canonical dissection

179

into each other~ may be realized by a homeomorphism. One can extend it to ]E/G when

one maps the associated disks correspondir~zly. If the au-~or:~rphiem ~ is~duces

a, then the homeomorphism of the plane which cover,s the homeomorphism of the surface

is a realization of ~. ~ae conditions that ~ permutes the boundary curves of rotations

of the same order and leaves fixed the curve around the base point are expressed

algebraically as

~.

~(S.) : L.S 1 L-1 1

I

~. i

l

[(IISi~[Ti,Ui]) = L(~Si~ [Ti,Ui] )SL-i for c, si = _+ 1. We shall attain e~n ~ to a given ~.

5,2 B I N A R Y

PRODUCTS

For the proof of the Debt-Nielsen tJ~eor~m we use some results on special

equations between elements of a free group. %]lis can be treated by a combinatorial group theoretic method which corresponds to ch~ngin~g 'geometrical' generators of fundamental groups of surfaces by bifurcations: the method of binary products. 5.2.1 Definition. Let S be a free group on the generators $1,$2, .... Let X1,X2,...,Xn be elements of S with s~i -1 Xi = Xi i Xi ' ci = + . 1, . i. : .1, .

,m < n.

Let XI~...,X n be symbols and let ~X = HX (X1,... ~Xn) be a word Jn the X~ 1 such that

each symbol X1,... ,Xm appears exactly once (either with exponent + 1 or - 1) m, e C { 1 , - 1 } . Then l e t

gi = XiX' gj = Xj, j ¢ i, Yi = XiX' Yj = Xj, j ~ i and ~V =

...y .... 1

(y.y-1)e

....

1

(d) If HX = ...XXz....X~...~ i > m, a @ {1,-1} let V i = XXi, Vj : Xj, j ~ i, Yi = XXi' Yj = Xj, j ¢ i,

Hy

""Yi""

(g-lgz

....

5.2.3 Defir~ition. Two binary products are called related if one may be converted into

the other by finitely many of the following processes:

(a) Renumbering the fir'st m and the last n-m generators• (b) Replacement of a factor by its inverse. (c) Bifurcations 5.2.2 (a-d).

The following properties are Ln~nediate. 5.2.4 Len~na. (a) The factors of related products generate the so,he subgroup of S.

(b) Related binary products have the same value Hx(X) in S. (c) Alternating products remain alternating.

[]

In order to describe the bifurcation process (and the processes 5.2.3 (a~b))

in geometric terms, we consider a surface with m perforations. Then the bounda_~y

curve of the disk which results from a dissection defines a binary product. It is

alternating when the surface is orientable~ but not in the non-orientable case.

If X.l is a closed curve of the dissection Which traverses the ith boundary once,

and if X is the curve of the dissection which follows X.z in the boundary path, ther~ X-1X.X is also a curve which traverses the ith boundary once, and i

... X(X-1X.X) ... is again a dissection, resulting from the first by a bifurcation z of the surface. The other processes may be visualized analogously (cf• 3.2). Thus: 5 . 2 . 5 Lenma. If a binary product stems from cutting a surface (with boundaries), then all related binary products likewise stem from cutting this surface.

Each. factor of a binary product has a lep~th as a word in the generators S i,

and we can speak of halves of factors. A factor X is called inessential when it

stands next to X -1 in ~X or is the identity of S. The process for finding generators

of a subgroup of a free group with the Nielsen property can also be used to prove the following theorem (cf. [Zieschang 1964] ). 5.2.6

Theorem.

properties :

For each binary product there is a related binary product with the

t81

(a) No more than half of ~ny essential factor (b) No essential

is cancelled by a neighbour.

factor has halves cancelled by both neighbours.

(c) The initial factor,

if essential,

loses less than a half by cancellation.

One obtains 5.2.6 (c) by treating an initial factor

lation as an factor

[]

which loses half by cancel-

which has halves cancelled on both sides.

5.2.7 Definition. A binsm%~ product which satisfies 5.2.6 (a) is called reduced, if itsatisfies

5.2.6 (b,c) also, then it has the Nielsen property.

Let the group S have free generators S1,...,S m and either T1,U~,...,T ,U or ± 2 g 2g V1,...,Vg respectively, and let H. = S1...S m ~[li,U i] or ft. = S1...S m V1...Vg

respectively. In order to avoid distinguishing the Sac cases cont£nual!y~ we write S in the generators H1,...,Hn, {H1,...,Hn;~.} is a binary product.

5 . 2 . 8 Theorem. I f

g.

{ X l , . . . , X n 6 f f X} is a binary product in S with

. XI1, . 1 < r i < m, i < m' , c. : ± 1 and fix(X) = ff (H) in S then n' . < n, X i. : XiHr% l

m' = m, n' = n and {X1,...,X;K X} is related to {H1,...,Hn;~,}. Proof. We remove all inessential elements of {Xl~...,X n, ;HX} and obtain a binary

product {X1,...,Xn,,;[ i} with n" _< n and m" = m' factors which appear once. This is converted into a related product with the Nielsen property, which we shall denote

by the same symbols. We remove all inessential elements from this product and again work to achieve the Nielsen property. This process terminates after a finite number of steps in a binary product without inessential elements, and with the Nielsen

property. In it, at least one letter remains of each factor. Let X[ 1 be the first factor of the form LiSI.L[.I~ ~ in Hk" We now write simply r in place of r i. S r is not cancelled, so s = + 1. If K is the part of H X' which stands before X -l+.

then we can

" by finitely many of the processes 5.2.2 (b), convert ff~ into (KLiSr [-1K 1 -1) ... = fix as a result of which X i goes into X' : (KL~S L71K-1). It is clear that S

not cancelled

in ~.

1

1

r

1

S i n c e ff~(X) : Hx(X) , t h e n ff~(X) : (KLiSpL?llK-1) . . .

form (81. .. . S r _ l S r S r _ l . . . 8 1 1 )

Sl...8r_lSr+l...8

m....

r

is also

has t h e

I f we l e a v e Xi o u t o f Hi and

S r out of H, then we obtaLn binaz%~ products {X1,...,Xi_l~Xi+l, ...,Xn,,;~_X} and

{S 1 , • .. ,Sr_ 1 , Sr+ 1 , • . . ,Hm+ 1 , • . . Hn;ff _.:~} which have the same value in S nmnely S1...Sr_ISr+I...Sm~q[Ti,U i] or S1...Sr_]Sr+I...SV~...V g2 respectively. The

hypotheses of the theorem are still satisfied, though m and m' are reduced by one. By a series of steps we can remove all factors from HX which appear once. If one

abe!ianizes and computes nod 2 then one sees that no factor appears only once in

ft, either (i.e. m" = m). Because of the Nielsen property we can also conclude that

the first factor equals S 1 . By induction, we can go to a related binary product in which the first m factors equal 81,...,S m.

182

We have therefore reduced Theorem 5.2.8 to the following situation:

{X1, ...,Xn, ;fX } is a binary product in which each factor appears twice, and H

is

a product of conmutators [Ti,Ui] or squares V21 respectively. To be sure, generators

S i may still appear in the words X.. We suppose that {X1,.•. ,Xn, ;EX} has the Nielsen property and no inessential elements appear,

and first deal with the "non-orientable"

case fix(X) = ft.(V) = V2...V 2, By renumber,ins, let Xll be the first factor in H)(, so I n that X11 2 2 2 2 for i < g~ where L is the part of X~ 1 - = V1...Vi_IViL or X ±1 1 : V1...V~L ~ + i. cancelled Ln fiX" Here g(L) ~ 2(i-1) or 2(_-~) ~ SLnce Xll appears again in HX~

the second case cannot occur. In the other case at most V. can remain from the i

2 .Vl-J 2 ~V., greater front half VI"'t and this is so because of the Nielsen property also. Therefore, the factor following Xll is again X~ 1, and we have X11 : VI""Vi-IVi(VI"'" 2 2 2 V i-1 2 )-1 . As above it follows f~sm i - 1 > O~ because of the

Nielsen property, that n' < n. We now have HX = XI_+2HX,with

2 V i-1 2 V i+1"'" 2 V n" 2 If we now apply the same process to {X2,...,Xn, ;H~} : V1... , '

f~(X)

then n' < n gives a contradiction. But it follows by induction from n' = n that the original binary product is related to {VI~...,Vn~H } . In the "orientable" case Zx(X) = T"U1 T-1U-1~ i - "'T U T-1U -1 with g : n/2. Let X 1 ggg g be the first factor in HX. Because of 5.2.6 (c), less than half of X 1 is cancelled. If L denotes the part of X 1 which is cancelled, we have the following four cases. i-1 j~l [Tj'Uj]TiL

5.2.9

X1

£(L) :~ooi of zi:e Delta-Nielsen theormr~.

We prove it first for closed surfaces ~ next for surfaces with ~]undary, amid finally i~] section 5.8 for pla~nas~ groups wh:ic]t have com]:~act fuz~dm~ental domain a~d do

not contain reflections.

Let G :

or respectively.

In order to avoid

case distinctions we write G = . As before: let G be the free group on the generators H1,...~H n. We thirx of the binJry Fr.cduct (H1,...,Hn;H p

realised

by a system of segments n I ~... ~nn a~:d a s~;:~pl~ clo~ed c~'~'ve i].(n) in the plane neL

fop tha a~ove presentation of @ (of. ~he ~poof of i% eorem 4.7.1)~ op by a ca~nonical curve system~ on the closed sLy-face 16/G. Cu:r along -w'e s ~ p i e

closed curves

n l , . . . , r l n c o n v e r t E/G i n t o a face wiN~ bouncary ]i.(r/). 5.6.1 ~[heorem. Each automorphism a of G is induced by au automorphism ~ of G with

^aft

: ~I ± 1 L - 1 "

Proof. let ~ be am endomorphism of G which indu. i=l i j=l 9

Nc~ we will represent each conjugacy class by a word of a special type, and it turns out that this word is well defined up to a cyclic permutation. We again denote the defining relation by H .

5.10.1 Definition. A word W is called a representative of a conjugacy class of G

if it has the following properties: (a) W is cyclically reduced.

(b) W does not contain a subword which comprises more than half of the defining relation ~ or its inverse; both are considered as cyclic words. (c) If W contair~ a subword which comprises half of
0 the t I or the e-th power of the first v 1. 5.10.2 Theorem on the sokution of the conjugacy p~mblem. The notation is as above

(a) Each conjugacy class contains an element that can be represented by a word with

206 the properties above. Let the length of the defining relation be greater than 7. (b) If two words with the properties above represent the same conjugacy class then they coincide up to a cyclic permutation Proof. (a) is trivial. If the defining relation has a length greater 7 then this follows from Dehn's solution of the conjugaey problem~ see 4.9.3. The proof there was oi~y for discontinuous g~)ups wit~ compact fundamental domain, not for the fundamental 87~oups of surfaces with boundary. But this case can easily be reduced to the one considered as follows: For some k -> 9, we i_utroduce the additional k relations si~ i 2 21 2 1. There is then a cyclically reduced subword N(H)

of M with W-1NW(H) = M(H) and w-l(h)~(h)W(h)V(h)

= 1 in G. We have a]i?eady seen

how we can apply the solution of the conjugacy problem to G, also for the case where boundaries appear. ~

can therefore be required to have the properties

3.5.13-15, so t~at it is the same cyclic word as V. However~ we can ass~me that N

contains no more than a half of a relation when regarded as a cyclic ~ r d ,

that it satisfies 3.5.15. But then the same is also true for ~

a=nd

because among the

words 3.5.5-12, only the words VIV 1 and VI...VgV 1. .Vg,S odd, come into consideration for V(H), and these bound H~bius strips, hence are excluded by assumption.

We shall now modify the isotopy H so as to leave the point P fixed. Since

is null-homotopic there are finitely many processes K1,...,K r of the type 2.4.2 (a) or (b) which convert ~ into the constant path. If r = 0 there is n©thin~ to

prove. We assume then that the desired modification can be made for r < k, and next consider r = k.

Let K 1 be the removal of a spur a~ -1

from ~, and suppose [tl,tl] is the sime

interval of I during which P traverses this spur. There is an isotopy Gt, 0 < t _< 1 with G t = G

= id F for 0 _< t _< t I and G t : Gt2 for t 2 ~ t -< 1, so that

HtGt(P) = Htl(P) for t E [tl,t2] , i.e. for eaeh point of time t E [t~,t 2] we hold P back at its initial position Htl(P) in the interval [tl,tg]. The existence of this isotopy can be seen intuitively by considering the part of F x [tl,t2] over a and a "small" neighbourhood around it. ~1 is

the old path of P in F x [t~t2], given by

t ~ (Ht(P),t), ~2 the new path: t ~ (H t (P),t). The isotopy GH still car~ies y into •

•

1

6, but a may be carrled Into the constant path by K2,...,K r, hence by a smaller

number of combinatorial isotopies.

213

One follows the reverse procedure when K 1 is the insertion of a spur. Then ~2

is carried into ~1"

I ~1

~ .

~1 ~ ~

[o

One proceeds analogously when K 1 is the replacement of psrt of the boundary

path of a face by another which forms a triangle Jn conj~nction with the first.

Here again only points in a small neig~bourhood of the triangle need to be m~ved. The induction hypothesis then yields an isotopy which leaves P f ~ e d and carries V into ~.

[]

The assumption that The curves boua-~d neither disks nor H~bius strips is

necessary. For disks we have seen this in 5.11.3; for M6bius strips we will show it now. Again we use the isotopy concept from algebraic topology.

5.12.4 Theorem. Let F be a surface and let P be the basepoint on F and (for instance,

~ that on S 1

the complex number I), let the image of fo: ($1'~) ~ (F,P) bound a

M~bius strip in F. We assume that fo(S 1) is disjoint from the boundary of F. Then

there is a isotopy ft: ($1'~) ~ (F,P) such that fl is an embedding, which is not iso-

topic to foWi ~

basepoint fixed. (The images ft(S 1 ) are disjoint from the boundary

of F.) Proof. Diagram (a) shows a slightly larger MSbius strip than that bounded by

fo(S1). The embedding fl shown in the diagram is isotopic to fo' by an isotopy which

chan@es omly the vertical coordinate in the diagram.

To show that there is no isotopy that leads from fo to fl and keeps the base

point fixed all the time, we examine two cases. Suppose first that F is a projective plane. Then fo and fl bound disks with opposite orientations and we apply 5.11.3. Suppose now that F is not a projective plane. Then every multiple of fo is

not trivial in the fundamental group of F. Lifting to the universal cover we see

that each component of the inverse image of the MSbius strip is an infinite strip.

Let h be the covering transformation correspondirz to a generator of the fundamental group of the Y~bius strip, and let f~' fl'" ([0,1],0) + (F',P') be liftings of f ,f~; here P' denotes the basepoint of F'. Suppose we had an isotopy o 1• ft: (S ,~) - (F,P). Then this would lift to an isotopy

f~: ([0,1],

0,1) ~ ( ( F ' \ h ( P ' ) ) , P ' , h 2 ( p ' ) ) ,

see diagram (b). In particular, the simple closed curve obtained by going first

along f8([0,1]) a~id then back along f~([0,1]) would be nullhomotopic, and would

214

therefore bound a disk in F'\ h(P'); diagram (b) shows that this is not the case.

p'

~

f! O

"h(P')

fl fi S. (a)

)

fig. (b)

[Baer 1928] has proved 8.11.2 for closed orienkable surfaces~

his approach is similar to that used by us in section 5.14. A combinatorial proof of the same result is in [Goeritz 1933]~ it is similar to our approach in section 5.11. The refinement 5.12.1 has first been fom~d by Epstein and was published in

[Epstein 1966]. Slightly later and only for the 'most interesting' cases, but not

knowing the Epstein success the result was found by Zieschang and published first in [ZVC 1970] ; this proof is given here. The 'counterexample'

5.12.4 was the result

of discussions of 'proofs' of a statement like 5.12.1 without the restriction about M~bius strips~ it was published in [Epsteir~-Zieschang

5,13

INNER AUTOMORPHISMS AND

1966].

ISOTOPIES

By iterated application of 5.11.1 we obtain an isotopy theorem for canonical

dissections and deduce from it that homeomorphisms

homotopic to the identity are

isotopic to the identity. This generalizes the result from [Baer 1928] from closed orientable surfaces to arbitrary ones. Moreover, using Epstein's refinement

5.12.1 of the Baer theorem on simple closed curves we prove the refinement that

a homotopy with fixed basepoint can be replaced by an isotopy which fixes the basepoint. 5.13.1 Theorem. If a homeomorphism h of the surface F induces an inner automorphism

of the fundamental group G, then h is a combinatorial isotopy of F (in the semi-

215

linear case there is an isotopy H of F from the identity to h). Proof. Let ~ be a canonical curve system with basepoint P. The first curve n! of E is ho~topic to h(n 1) since h induces ~n in~er automor~hism, so by Theorem 5.11.1 there is m~ isotopy (combinatorial isotopy) H 1 which maps h(n I) on to n I arld carries h(P) into P. By transporting P along Hlh(n 1) if necessary we can assume that corresponding

curves of E and Hlh(E) are ho~)topic after fixing the basepoint. (Cf. the proof of

Theorem 5.12.1{ there it was shown that, possibly after pushing P along ¥, the path of P was null-homotopic, and this is precisely the propert-y needed here. ) If one

applies the method of proof for Theorem 5.32.1 to fk~e curve system then the result is a series of isotopies H1, H2, ...

We finally obtain an isotopy

H n with Hi...Hlh(n j) = n., j = 1,...,i.

}] of F which maps h(E) onto Z. 19~ is a homeomorphism

of F which is the identity on F.. If we remove the aRnuli which aaoe bounded by the curves ~l,...,~m of E and then cut F along r, we obtain a disk. Hh induces a

homea~mrphism of the dis]< which is the identity on the boundary. By the Alexander

lemmm 5.13.3 below, such a homeomorphism is a combinatorial isotopy. Likewise, the

homeomorphism of the annulus bounded by ~I''" "'~m which is induced by [rn is a combinatorial isotopy. This also follows from Lemma 5.13.3.

[]

5.13.2 Theorem. If a homeomorphism h of F leaves the basepoint P of the fundamental

group G fixed, and if it induces the identity on @, then h is isotopically deformable into the identity in such a way that P is fixed throughout the deformation.

Proof. One applies ~neorem 5.12.1 to nl, which does not bound a H~bius strip,and %hen proceeds as above.

[]

5.13.3 Lem~na (Al~xander-Tietze deformltion theorem). A homeomorphism h of a disk

which i~ the identity on the boundary may be deformed into the identity by an isotopy which does not move the points of the boundary.

Proof. Let the disk be the unit circle {(x~y)Ix 2 + y2 ~ 1} in the euclidean plane 2{2 , and let h be the homeomorphism. Assume

h(O,O) = (0,0). Let G t be the dilatation

(x,y) ~ (tx,ty) of ]{2 and let H be the holreomorphism which equals h on the unit disk

a~nd is the identity elsewhere. If now H

is the identity a~nd H t = GtHG~ 1

for 0 < t _< 1, then this defines an isotopy of h into the identity. - If h does

not fix (0,0) we move first h(O,O) back to (0,0) by an isotopy of the disk which is the identity on the boundary and then apply the above ar~nent.

D

216

5.13.4 Remark. ~uis isotopy is s~milinear if we take a triangle in place of the

circle, with the same midpoint. An analogous proof may obviously be carried out in higher dimensions.

For closed orientable surfaces theorem 5.13.1 was first proved in [Baer 1928].

A combinatorial proof similar to our is in [Goeritz 1933, 1933']. For arbitrary closed surfaces it is in [Br~del 1935] ~s~d [Mangler 1939]. The general version

and its proof is from [Zieschang 1966]. ']%~erefinement 5.13.2 was first proved

in [Epstein 1966]. The Alexander-Tietze defo~mmtion theorem was first proved in ~ietze 1914 ]. The proof here is from [Alexm~der 1923]; it is now standard in

books on algebraic or geometric topology.

In section 5.14 we give a generalization of the Baer theormn and in 5.15 we

will discuss the literature of further developments connected with it. Exercises: 5.14-16

5,1L~ THE BAER THEOREM FOR PLANAR DISCONTINUOUS GROUPS In this section we generalize the Baer theorem to all finitely generated

planar discontinuous groups that contain only orientation preserving transformations. (The last assumption can be dropped, but we will make it for sinplicity.) Later,

in 5.16, we will use the main theorem 5.14.1 to prove relative versions of the Baer theorem. 5.14.1 Baer Theorem. Let G be a finitely generated discontinuous group of orientation

preserving mappir~s of E which is not cyclic. Let ~ : E ~ E

be an orientation pre-

serving homeomorphism such that 5.14.2

~ - 1 g % : g for all g 6 G

Then there is an isotopy Ct: E ÷ E 5.14.3

w/th @o = t, ¢1 = i ~

and

¢~1 get = g for all g 6 G and t 6 [0,1].

It is convenient to imagine mmppings ~nd isotopies as usually done in algebraic

topology. Before we make some remsmks to the theorem and start the proof we introduce some useful notation.

217

5.1'~.4 Notation. For O we take a canonical ftmdamental domain F as described in

4.4, 4.11; the boundary of F has the form 5o14.5 ~ j 1 1 . . . S m ~ m l p i [ ] p 1 1 . where each [] denotes an

"

.pq[]

-1 , -1 -1 , , -1 -1 , P q~lPl ~I ~l"''~gPg ~g Pg

'end' of F. For s~mplicity, we put

~1 = Ol''"'~m : ~m'~m+l = P l " " ' ~ m + q

~ Pg'am+q+l : TI'

~m÷q+2 = Pl'"" 'am÷q+2g = ~g' and similarly

for the curves with a prime. To

each pair Of directed boundary edges ai,a'i there corresponds an element of G sending ai to

i' we denote it by a i. By n =: m+q+2g

we denote the n~nber of genera-

tors. If we use the special notation for ~i' for instance

~j,then we use the

canonical notation si etc. The group G has the canonical presentation

kI k m q g 5.14.6 O = "

By N we denote the net obtained from F by the action of G. G operates simply

transitively on the faces of N. We take a point p. from the ]x~terior of F and join it wit/~ aiP* by a broken line Yi (i = 1~... ,n) which crosses ~i once; ~i sb~ll not pass fixed points of transformations of G. Let C = {~iii : 1,... ,n}. GC defines a

net N* in ]{, dual to N. For g ~ G the edge

gYi curs one arid only one edge

bmlflLne from N rand this piece is G-equivale~t to el" 0, then p ~ n.)

Points of ]~ that are mapped to the same point of S belong to the some fiber. This presentation of G arid this type of net N will be used~ if G a~m.its a proper decomposition as a free product with ~m~algamation, see

5.14.17.

218

5.14.8 Remarks. (a) 5.14.2 expresses that % is fiber preserving and 5.14.3 describes

a fiber isotopy.

(b) From 5.14.2 it follows that ~ fixes all fixed points of non-trivial eler~ents of G

(c) If @ does not contain elements of 'finite order (i.e. m : O)

then 5.14.1 is a

reformulation of the Baer theorem 5.13.1. Our proof is quite close to the original

one in [Baer 1928].

The proof of 5.14.1 will be finished in 5.14.33. First we shall prove some

lenmata.

Fr~)m 5.14.8 (b) it follows that t(C) does not contain any point fixed by a

non-trivial mapping. So general position arguments for the projection ~ ~ - 1

show

that ~ cmn be deformed by a fiber isotopy such that 5.14.9 (a) ~ C meets N only in a finite number of points. (b) ~C o H - l p o : ~

This allows us to define for each element ty of }C a description, i.e. a

word in the generators from 5.14.6 and their inverses as letters,@~ich reflects how ~y crosses N.

5.14.10. Let Wi(t) denote the word correspondi~s to }(yi ). The lensth di(t) of

Wi(~) is the number of letters occurring in Wi(~). Let d(~) be the sun of the d.(~).l If a word Wi(~) contains a pair aa'aTEl l (s = -+ 1), we say that ~ allows a free can-

cellation.

5.14.11 L ~ .

If a free cancellation is possible for ~ then ~ can be defornrzed by

a fiber isotopy to a mapping ~1 such that d(t 1) < d(t).

which also has the properties

5.14.2

and 5.14.9

Proof. Repeat the argument from (1) in the proof 5.11.5 of theoreml 5.11.1. See the figure.

[]

219

U

5.14.12 Lemma. If all Wi(~) start with the same letter and end with its inverse,

can be deformed by a fiber isotopy to a mapping ~1 with the properties 5.14.2 and 5.1~.9 such that d(@l) < d(~). Proof. The ass~r~ption says that all curves in t(0C) which start or end at t(pe) go to the same edge of }~ or c o ~

from it resp. It follows as in 5.14.11 that

there exists a fiber isotopy which moves @ ( p )

to the other side of the segment

and which dkmdnishes d(%).

D

5.14.13 Definition. Let us call a mapping ~ cyclically reduced, ~

free

cancellations are impossible and the situation 5.14.12 does not occur. 5.14.14 Len~na. Let ¢ be cyclically reduced. Then Wi(%) , 1 ~ i S n, does not contain

a subword s~s~(j = 1 .... m) or r~r~(j = 1 ..... q), e =± 1. JJ

'

33

Proof. We assume that in Wi( ~ ) a letter is followed by itself and consider the -' which image on S = 3E/O, where the situation looks as in the figure. The arc ~i

goes inside produces an infinite spiral, because by ass~amption it c ~ n o t H@(p,) and does not allow free cancellations.

end in []

220

Yi )

5.14.15 Lem~na. Let ~1'" "" '~h be successive curves from the star at pc ~ and -

Bh+l~...

~p the rest.

We a s ~ m e t ~ t ¢ is cyclically reduced. Let 6 be a part of

one of the curves ~¢(yi ), 1

X:~ ~ Z 2

Ibl) from its image b + i P / ~

2 is

log(-b+i t2V~-b2 -t -b+i t2V~-b2 +t b+i ~

-t

b+iVt2-b 2 +t

As t - ~ this expression converges to O.

6.3.7 Definition. The non-euclidean arc element is given by Idzl ds : - Y where z : x + iy. 1%le continuously differentiable functions which leave this element invariant are exactly the motions of non-euclidean geometry. The non-euclidean sur-

face element dw = dxdy 2 Y is likewise invariant under non-euclidean motions. By a non-euclidean polygon we

mean a region in the non-euclidean plane bounded by non-euclidean line segments. Using the Green formula or the Gauss-Bonnet formula one obtains

6.3.8 Theorem. The non-euclidean area of a non-euclidean polygon with n vertices

and internal angles ~l,...,an (n-2)~r-

is

(el + "'" + a ).

Proof is exercise E 6.6. Furthermore, vertices may also lie on the real axis, in which case the corres-

ponding sides will be half lines with the same end, Then the angle is reckoned to be O.

For a detailed exposition of the material in this paragraph see textbooks on

function theory, especially [Lehner 1964

Exercises: E 6.1-6, E 6.13.

], [Siegel, 1964 T ] ~ [Zieschang, 1980].

247

6,4 6.4.1

PLANAR DISCONTINUOUS GROUPS

If a group B operates on a topological space R then B

can be given a topo-

logy which reflects the fact that "small" cha~ges in the mappings displace the image

points only slightly. A topology which is frequently copsidered has a subbasis

consisting of the sets {x C B : xU c V} where the U are the compact, and the V the

open, subsets of R. Obviously subgroups receive the induced topology. If B consists

of motions of the euclidean or non-eucl~dean plaJ~e then one obtains the topology naturally associated with the parameters a, b, c, d. 6.4.2 Definition. A group B of homeomorphis~

is called discontinuous when there is

no point P E R for which the set {xP: x E B } has an accumulation point. In particular, a point can only be a fixed point finitely often. 6.4.3 Theorem. A subgroup of the motions of the euclidean or non-euclidean plane is

discrete if and only if it operates discontinuously on the plane. The proof is an exercise.

[]

The following theoz~m of [Nielsen 1940] may also be mentioned, see [Siegel 1950,

1964' pg. 39].

6.4.4 Theorem. If a non-commutative

subgroup of the motions of the non-euclidean

plane contains only hyperbolic transformations, then it is discrete.

[]

We say that a discontinuous group has compact fundamental domain when the images

of some compact subset cover the whole space. This is equivalent to saying that the factor space modulo the group is comvact.

6.4.6 Theorem. If a discontinuous group G of motions of the euclidean or non-euclidean

plane has compact f u r ~ e n t a l

domain then there is a net in the plane, the edges and

faces of which are line segments and convex polygons of the corresponding geometry. The requirement that the group has

compact fundamental domain is unmecessarily

strong. The idea of the proof may be carried over to the general case, and one like-

wise obtains a net, the faces of which may however have half lines and possibly

pieces of ~ne real axis in ~ne~r bomndaries, indeed this must happen if the fundamental domain is not compact, cf. [Siegel 1964', p. 42].

248

Proof. Since the group is discontinuous there is a point P E ~

or C respectively

which is not fixed under any transformation in 0 except the identity. Px again denotes the image of P under application of x E O. For each x C @, x # 1 we take the closed half-plane consisting of the points of the plane which are no further from P than from Px. (Geometrical terms refer to the appropriate geometry.) This is the half-plane defined by the perpendicular bisector of the line from P to Px and containing P. Let the intersection of all these half-planes be Z it is the set of points which are at least as far from each of the other points Px (x # 1) as they are from P. If Q c F then Q is nearer to P than any other Qx (x ¢ 1). If two points from the same equivalence class are both in F then they must lie on the boundary of F. Therefore F is a fundamental domain and contains the points from each equivalence class which are of minimal distance from P. Now there is a circle around P which contains a point from each class, hence, it contains F;(this is where the compactness of the fundamental domain comes in). If we then take the equivalents of the point P which are at a distance of more than two radii from P, then the corresponding circles do not meet the one centered on P. But since G is discontinuous, we have omitted only finitely many transformations, i.e. F is the intersection of finitely many half-planes and compact, thus a convex polygon. If one constructs the corresponding polygon for each point equivalent to P, then one obtains a planar net on which the group acts. This is a consequence of the fact that F contains exactly the points of minimal distance from P from each equivalence class.

[]

We have shown in this way that each discontinuous group of motions of the plane with compact fundamental domain carries a planar net into itself, and thus is of the type treated algebraically in Chapter 4. Conversely, it also holds that each of these groups may be realized by motions: 6.4.7 Theorem. A group which is given by generators and defining relations of the

type A or B in Theorem 4.8 is realizable as a discontinuous group of motions on the sphere (>), the

euclidean (=) or non-euclidean plane (< 2

E i=l

. i

or in case B

E E i=l j:l

> 0 v 2 mE (1-~-7.)1 + Zq i=l

m

IN.

. .) + 4g - 4 + 2q ~3

1 + ZI (1 _ )h.. i=l j=l 13

2g - 4 + 2q

249

Proof. First we choose a polygon, on the sphere, euclidean plane or non-euclidean

plane respectively, the sides of which are denoted by s~mbols as in Theorem 4.3.6,

so that it takes the form 4.3.7 or 4.3.8. Segments denoted by the same Greek letters

and in~ices will have the same length, the segments denoted by gi and g!i will meet 2~ in the-angle ~ii those denoted by ¥k,i and ¥k,i+l in the angle hki~T, the sum of tAe angles between nl and Yk and between Yk,mk+ 1 and nk will be ~, and the sum of the

remainin~ angles will be 2~. It is known

that such polygons exist on the sphere

and in the euclidean plane; moreover they correspond to well-known groups. In the non-euclidean plane one proves the existence of such a polygon by a continuity

argument. (Small polygons ape "neaPly euclidean", hence can have angle sum arbitrarily close to the euclidean value, while large polygons have angle sum arbitrarily close to O. %~qen by continuit%, we can get storyvalue in betseen. )

In Theorem 4.7.1 we began with the groups, and constructed planar nets on which

they act. We take the dual net to that deseribed in the theorem. Here the group acts simply transitively on the faces, the boundaries of which may be naturally

~m£tten in 19~e form 4.3.7 or, 4.8.8. If we replace the faces of the net by the polyson and its labelling then we obtain a Rienmrm surface, since for each point we can find a nei~ourhoodholomo~hic

to the ~±nit circle. For interioPpoints of the polygon

this is trivial, for boundary points which are not vertices it follows from the fact

that segments with the same label have equal length; and the angle conditions take care of the vertices. Moreover, the Riemar~l surface is open and simply connected,

and G operates conformally on it. Then on the basis of the Riemann mapping theorem G

may be realized by a discontinuous group of motions of euclidean or non-euclidean geometry.

D

6.4.8 Remarks. a) I t a l s o follows from the inequalities for the angle sum of a con-

vex polygon that one can determine the plane on which the group operates analytically from the inequalities in tAe theorem. b) It is natural, possible and frequently done in the literature (though often

carried out incorrectly), to construct the net in the pla~ne by successively laying down polygons ; for locally congruent polygons can be laid next to ead~ other as

required, and generatin~ motions defined as a result, which satisfy the defining

relations. Beginning with a polygon, this step-by-step process gives us a holomorphic mapping of the net constructed in the euclidean or non-euclidean plane, and in fact

a covering without branching. Since the plane is simply co~unected and since one can

reach every point in the plane by a suitable chain of congruent polygons, the map-

ping turns out to be "onto"

amd our Riemanp. surface is holo~orphic to the plane.

The discontinuouts group therefore consists of motions of the geometry.

250 c) The Riemann-Hurwitz

4.14.23 :is immediate in this context. It simply

formula

compa:pes t h e a r e a s o f fts~damenLal domains f o r g r o u p and s u b g r o u p .

Exercises: E 6.7-9

6,5

ON THE HODULAR PROBLEH In tk£s section questions concerning ~he complex-analytic

structure of Riemarm

surfaces arS plantar discontinuous groups are discussed. From dne Riemann mapping

theorem [Beh~J d(qk) , an inequality d(~:)

d ( n ~ ) mus~ hold, say d(~ 1) a d(nkl)-

J

Consequently d(qkle 2) d(e2). But then the curve -1 ~2 nk2 is shorter than nk, and hence

nk2 by (Ek) it must be homologous to a sum in nl,...,nk_ 1. Computing homologically, ~lnk2 and ~ differ only by a sum of ql,...~nk_ 1 and thus o;lnk2 , like

w~ is homolo-

gically independent of nl~...,nk_ 1 for i _> k. ~nerefore we must have d(nk2) > d(~ 2) and nk may be replaced by the shorter nkle2 and (Ek) cannot hold. %-bus if we take as hi+ 1, (Ei+ 1) will hold.

[]

Proof of Theorem 6.7.3. By means of Lemma 6.7.8 we finally find a geodetic dissection which contains two closed geodesics - n I aTJ K here. As a result~ the sum of two adjacent angles cannot exceed ~, and the dissection is doubly convex.

[]

6.7.9 Lemma. Let Z be a doubly-convex geodetic dissection of R a n d F a fundamental

domain of F in ~ the boundary of which is mapped onto Z. Assume that F n t(F), t c F, is the segment PQ.

(a) Then there is a point A E P-~such that t-I(A),A,t(A) are on one line and this is

259

the axis of t, see figure. (b) If tl~t 2 are two generators defined by F and if they correspond to curves on R with algebraic intersection number ± 1 then the axes of t I and t 2 cross each other in F.

Proof. For the s~Kbols used see figure • 'l~e doubly-convexity

of Z implies that the

angles a and ~ do not exceed ~. Now we take a point X 6 pQ and consider the angle

bet~een t-l(x~x and Xt(X). It varies continuously from ~_< ~ to 2~-B->~. Hence there is an A as required. If, for a linear fractional transfo~n-ration t, the points A, -1 t (A) ~ t(A) are on one i~ne then the ~_w~sfo~,mtion is hyper~bnlic a~d rLe line

equals the axis of t; for a proof see [Zieschas] S 1980, 14.17] or E 6.13. Now (b) is a

direct consequence of (a).

t

(

~

I

~

~-~t-l(Q)

t(A t(F)

]

F

t(P) 6.7.10 Le~ma. Let Z be a doT~bly-convex geodetic dissection. Then each point of the

surface can serve as the basepoint of a geodetic dissection dual to Z or isotopic to Z respectively.

Proof. We again consider the universal coverir O. Let ~

be a closed arc on J, with ~

c U(a,~), which

293

has a as an interior point~ see the figure. Let 6 : d ( a , J k L 1 ) .

We now show:

(1) Two points x,y ~ D N U(a~6) may be connected by a path in D ~ U(a~a). To prove this let F

= ~U(a,~) = {x 6 ~ 2 1 d ( a , x )

= s}. Also, let ~ be a path in

U(a,6) which connects x and y. Then m[]

Fs

~0JXL i.e. x ~ y inlR2\ (F

:~

1 :~

U J \ LI).

Since x,y 6 D, x ~ y also holds i n l R 2 \ J . We have that J @ (F is path connected. By 7.1.5, x - y in]R2\ ( F

U J \ L 1) = J \ L 1 s U J X L 1 U J) : I ~ 2 \ ( F ~ U J) • 'This

~mplies that x and y cmn be connected by a path in D n U(a,e), i.e. (1) holds. %'&hen D U U = D is compact, the proposition follows from (1) m~d 7.2.4. If D is the infinite component of I~2\ J, we choose a circle K so large that J lies in the open disk determined by K. Let a : d(J,K) ; we have K c D. The region Do between K and J is compact, so that Do is ulc. l]~en, corresponding to the definition 7.2.2, let 6(c) > 0 belong to ~ > O. We extend this to D by taking 6' = rain {6(s),a} for the given ~ > O.

D

Analogously~ one shows 7.2.7 P~oposition. The complemen~ endpoints .

of a s i ~ l e

arc L in S 2 or ~ 2 is ic at its

294

This does not hold for interior points of L. as the following sketch slnows:

( For this section see [Newman 1951, VI ~ 4].

7,3

CONSTRUCTION OF A CROSSCUT In this section we fLnd a way to reach a point x on the b o u n d ~ y of a Jordan

domain by means of a curve which lies in the interior of the domain except at its

endpoimt x.

7.3.1 Definition. L~t D be a domai~l in i[2 . (a) A simple curve y is called an end cut of D when an endpoint of y lies on the botunc~ry of D, but all other points lie inside D. (b) A simple curve i is called a crosscut of D when both endpoints of I lie on the

botundary of D, bur a ~

other po£nts lie inside D.

(c) A point x on the boundary of D is called accessible from D when x is the end-

pint

of an end cut.

295

(d) An arc ¥ is called quasilinear when 7 is the union of at most denumerably many segments and each point of ~ with the possible exception of the endpoints is either in the interior of one of these segments or in the ~x)undary of two of them. In

other words, each closed proper subarc of y consists of finitely m~ny segments. Accessibility of points is a topological property of the pair (D,~D). The following gives an ~ a m p l e of an inaccessible point:

D

y

accessible,

x

not accessible

x

7.3.2 Lena.

Let D

be a domain in R 2 or S 2 .

(a) The points of ~D accessible from D are dense in aD. (b) If a point a C ~D is accessible~

then it is accessible by quasilinear arcs. One

can arrange that the segments of the arcs are alternately parallel to the real and imaginary axes. (c) If a 6 ~D is accessible, and b 6 D, then D contains a ~ a s i l i n e a r end cut from b

to a.

(d) If a~b 6 ~D are accessible,

then D contains a quasilinear crosscut from a to b.

In (c) and (d) one can also arrange that all segments are parallel to the real or imaginary axes, Proof. (a) Suppose we have s>O and a 6 aD, x 6 U(a,~) @ D. Let y be either the seg-

ment Ix:a] or an x L a consisting of one segment parallel to the real axis and one parallel to the imaginary axis. The first point of 7, after x, which lies in aD is accessible and lies in U(a,e).

296

a

(b) If y is ~

end cut

to a~ then ¥\{a} may be covered by a system of circular, disks

~i~id~ lie in D. Inside them, one car~ alter y into the desired form. (c) Let 7 be an endcut x ED

of D to the point a. Let the other endpoint of y be x. 'Fnen

and x may be connected to b by a s ~ p l e

ar,c ~. Now let y be the first point

of y~ after a~ which lies on h. Let Y1 be the subarc of ¥ between a and y~ A1 the sukmrc of A between y and b. Then ~1 u A1 is an endcut from b to a. a

x (d) follows analogously.

[]

Let D be a domain in S 2 or IR2 . If

7.3.3 Proposition.

D is lc at a 6 $D~

then a is accessible from D.

The proof uses 7 . 3 . 4 Lemma. Let a 61{ 2 and let A be the union of a sequence of segments wish

I~

Xn

:

l ~ l y n = ~,

[Xn~Yn]

Suppose that a l~es in the same path component

of ~ as the point b 6 A,b # a. Then there is a simple arc in A with en~points a and b.

297

1 1 Proof. Let C n : {x 61R21d(x,a) :2n d(b,a)} be the circle with radius ~ K d(b,a) and center a. Since only finitely many of the seg]nents [Xn,Y n] meet the region outside Cn, we use subdivision to arrange that outside of any C n two segments have at most an endpoint in ca~m.on~ a~nd a co,.on point of a segment a~nd a circle Cn :Ls aa endpoint of the segment. Now let A n be the part of A which lies outside ar on C n. Since a and b lie in the same path compone~it of A, there is a simple are J

n

in A

n

which leads from b to a point on C n . 7bus Jn N C n is a point. Since only finitely man}, segments of A lie outside C 1 , there is a simple arc dl which is the initial arc of iI~[initely mas~y of the Jn" l~t the endpoint of J1 equal b i, so that Jl G C 1 = b 1. Infinitely many of these ares have the same beginning J2 from b to a point b 2 6 C 2. We have that J2*\\ {b2} is disjoint from C 2. Thus for each n we j* * j* j* find a simple arc n in A with midpoints b mid bn : Cn O Jn" moreover n c n+!" Letb

O

=b.

There is a homeomorphism ~ of the subarc of J~n between bn and bn_ 1 onto the I 1 1

interval [ ~ ~

homeomorphism ~: ~: {a} U

U

n=!

J~

~] with ~ 0 so that any two points x,y 6 D @ U(a,6 n) can be 1 connected by a polygonal path in D N U(a,--). k~ 0 there is a crosscut

K in D from a to b with

(b) The crosscut K may be chosen to be quasi tinear; moreover, assumed parallel Proof.

d(K) < d(L) + ~. the segments can be

to the real or imaginary axes.

By 7.3.5 there are endcuts

Ya from a to a point p E D and Yb from b to

q 6 D. We may assume that d(Ya) : d(Tb) < e/2. We now show that p and q can be connected by a path in D N U(],: s/2).

1 To prove this let L ° be an arc in JXL sach ¢~aL: JXL ° c U(L, ~c). We will ~1ow amply E

7.1.5 to the points p~q and the compact sets L° J ~li(L~ ~) and J.

L ° U ~U(L, 7)s does not meet the ,m~,ve 7a U L U yb , and likewise J does not separate a ~nd b. Moreover~ J S(L ° U :gU(L~ 2 )} : J @ L ° = %

is path connected.

Thus by 7.1 5 there is a quasilinear simple path 71 from p to q which does not

meet J U (L ° u ~U(L, -~)). Since p,q 6 D N U(L, T) it follows that ~1 c D N U(L, -~). Then the desired path K may be put together fr@r~ ya ~ Yb and Y1 as in the proof of 7.3.2 (c).

For