Combinatorics of Train Tracks. (AM-125), Volume 125 9781400882458

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Annals of Mathematics Studies Number 125

Combinatorics of Train Tracks by

R. C. Penner with

J. L. Harer

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1992

Copyright © 1992 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, John Milnor, and Elias M. Stein

Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durabil­ ity of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

Library of Congress Cataloging-in-Publication Data Penner, R. C. 1956Combinatorics of train tracks / by R. C. Penner with J. L. Harer. p. cm. — (Annals of mathematics studies ; no. 125) Includes bibliographical references. ISBN 0-691-08764-4 (cloth) ISBN 0-691-02531-2 (pbk.) 1. Geodesics (Mathematics) 2. CW complexes. 3. Combinatorial analysis. I. Harer, J. (John), 1952- . II. Title. III. Title: Train tracks. IV. Series. QA649.P38 1991 511'.6—dc20 91-33559

Dedicated to my Father, Dr. Stanford S. Penner

CONTENTS

P r e f a c e ............................................................................................................. ix A c k n o w le d g e m e n ts .....................................................................................xi C h a p te r 1 T h e B asic T h e o r y .................................................................. 3 §1.1 Train Tracks ...................................................................................... 4 §1.2 Multiple Curves and Dehn’s Theorem ........................................ 10 §1.3 Recurrence and Transverse R e c u rre n c e ........................................ 18 §1.4 Genericity and Transverse Recurrence ........................................ 39 §1.5 Trainpaths and Transverse R e c u r re n c e ........................................ 60 §1.6 L am inations........................................................................................ 68 §1.7 Measured L a m in a tio n s .................................................................... 82 §1.8 Bounded Surfaces and Tracks with S t o p s .................................. 102 C h a p te r 2 C o m b in a to ria l E q u iv a le n c e .............................................. 115 §2.1 Splitting, Shifting, and C a rry in g .................................................. 116 §2.2 Equivalence of Birecurrent Train Tracks .................................. 124 §2.3 Splitting versus Shifting .............................................................. 127 §2.4 Equivalence versus C a rry in g .......................................................... 133 §2.5 Splitting and Efficiency.................................................................. 139 §2.6 The Standard Models .................................................................. 145 §2.7 Existence of the Standard M o d e l s .............................................. 154 §2.8 Uniqueness of the Standard Models .......................................... 160 C h a p te r 3 T h e S tru c tu re o f M C o ...................................................... 173 §3.1 The Topology of M C o and V C o .................................................. 174 §3.2 The Symplectic Structure of M C o .............................................. 182 §3.3 Topological Equivalence .............................................................. 188 §3.4 Duality and Tangential C o o r d in a te s .......................................... 191 E p i l o g u e ...................................................................................................... 204 A d d e n d u m The Action of Mapping Classes on M C o .......................... 210 B i b l i o g r a p h y .............................................................................................. 214

vii

PR EFA CE

We study here one aspect of the mathematics pioneered by William P. Thurston, namely, the rich combinatorial structure of the space of mea­ sured geodesic laminations in a fixed surface. One may think of this space as a natural completion of the collection of all IR+-weighted isotopy classes of essential simple closed curves in the surface, and it turns out to be a piecewise-linear manifold homeomorphic to a Euclidean space whose di­ mension depends upon the topological type of the surface. Roughly, a train track is a CW complex in the surface (together with extra structure), and appropriate train tracks correspond to charts on this manifold. More explicitly, a measure on a train track is an assignment of a nonnegative real number (satisfying certain conditions) to each edge of the underlying CW complex, and a measured train track gives rise to a corresponding mea­ sured geodesic lamination in the surface. We may thus explore the space of measured geodesic laminations by studying measured train tracks in the surface. Our techniques are almost always essentially combinatorial. For in­ stance, the first chapter (which includes the basic Thurston theory of train tracks) involves numerous analyses of separation properties of points in the circle at infinity of the hyperbolic plane. A combinatorially defined equiva­ lence relation on measured train tracks (implicit in [T2] and considered in [PI]) is systematically studied in the second chapter. Our investigation of this equivalence relation leads to a family of standard models (from [PI]) for the equivalence classes. In the third chapter, we rely on these standard models to study both the local and global structure of the space of measured geodesic laminations and derive many basic results of Thurston. Among these are certain global coordinates (from [PI]), called the Dehn-Thurston coordinates, on the space of measured geodesic laminations. There is thus much from [PI] which finds application in our approach. Because of the combinatorial nature of our arguments, there are few real pre-requisites to this volume. We take for granted basic differential topology (at the level, say, of [GP]), certain standard facts about surface topology (see [E]), and at least a working knowledge of elementary hyper­ bolic geometry at the level of [T2;§2]. (See [Be] for a more extensive treat­ ix

X

PREFACE

ment.) Furthermore, we assume certain elementary facts about Fuchsian groups, the treatm ent in [A] being much more than sufficient. Fundamen­ tal to us are two results: the Uniformization Theorem of Koebe, Klein, Poincare, and Ahlfors-Bers (which we use simply to guarantee that the surfaces we consider have the hyperbolic plane as universal cover), and the Nielsen Extension Theorem (which says that a homeomorphism of such a surface lifts and extends to a homeomorphism of the circle at infinity). A suitable reference for both of these basic results is [A]. The reader willing to take these results for granted should be able to tackle this volume with no further pre-requisites. Our discussion is completely independent (except for the Epilogue) of the parallel theory of measured foliations, but of course the reader familiar with Thuston’s work from [FLP] will greatly profit in insight. The action of homeomorphisms of a surface on curves in the surface extends to a continuous action of the mapping class group on the space of measured geodesic laminations. Train tracks are quite useful for study­ ing the dynamics both of this action and of a given homeomorphism on the surface itself. By and large, we treat only the “static” theory of mea­ sured geodesic laminations, in that we only rarely consider this action of the mapping class group. The two exceptions to this are in the Epilogue, where we survey aspects of the larger contexts of Riemann surfaces and surface automorphisms to which our investigations here are relevant, and the Addendum, which includes explicit formulas from [PI] for the action of the mapping class group on Dehn-Thurston coordinates.

In the bulk of the text, we treat surfaces with no boundary ( “of the first kind”), relegating the extension of results to the bounded case to re­ marks and exercises. For bounded surfaces, we discuss in the first chapter a generalization (from [PI]) of train tracks (called train tracks with stops), and the extension of our results to this setting is similarly left to remarks and exercises. The rationale for this is that these extensions are often straight-forward while the particulars of them may obfuscate the relevant material.

ACKNOWLEDGEMENTS

First and foremost, thanks go to Bill Thurston for being generous with his time during 1982-83 and 1984-85, when much of this work was done. Early drafts of certain sections (indeed most of §1.1, 1.3-1.5 and 3.2) repre­ sent joint work with John Harer, with whom there were many invigorating discussions. The proof of Theorem 1.4.4 was also kindly contributed by Nat Kuhn. There are intellectual debts to the lecture notes [Cb] and sub­ stantial such debts to [Tg], upon which §1.1 and 1.3 are roughly based. We thank Scott Wolpert for helpful comments on an early version of this manuscript and Max Bauer for clarifications of various arguments in the first chapter. M att Grayson, Steve Kerckhoff, and especially Lee Mosher made many valuable comments. It is a pleasure to finally acknowledge the support of the National Science Foundation as well as the warm hospitality of Insitut Mittag-Leffler during 1983-84 and Stanford University, especially Jim Milgram, during Spring, 1991.

COMBINATORICS OF TRAIN TRACKS

CHAPTER 1

TH E BA SIC THEORY

We begin with the basic definitions and ideas. Much of the ma­ terial of this chapter is due to Thurston. Our treatm ent includes new results as well as new proofs of facts from [Tl], [T2] and [Tg]. We define train tracks and introduce the notion of a trainpath on a train track, motivate the notion of transverse measure on a train track by considering a sense in which train tracks carry curves, and recall Dehn’s parametrization of curves in surfaces. The notions of recurrence, transverse recurrence, and tangential measure are introduced, and we provide several equivalent defi­ nitions. Generic train tracks (to which we restrict attention in later chapters) are introduced, and a certain geometric condition is shown to be equivalent to transverse recurrence in this setting. The importance of transverse recurrence as a technical condition is then highlighted by several results on trainpaths in transversely recurrent tracks. Laminations are defined, their elementary prop­ erties are explored, and an explicit construction of a measured geodesic lamination from a measured transversely recurrent train track is given. Surfaces with boundary are then considered, and the foregoing theory is seen to apply with essentially no modifi­ cations; we also introduce a relative notion of train tracks, called train tracks with stops (which arose in [PI]), and explore the basic theory in this setting.

3

§1.1 T R A IN TR A C K S

Let Fg be a closed, smooth, oriented surface of genus g with s > 0 distinguished points, whose union we denote A. We will usually regard A as deleted from the surface and define Fg = Fg —A so that each point of A gives rise to a cusp (or puncture) of Fg. When the topological type of the surface Fg is fixed or not important, we may call it simply F. Suppose that r C F is a finite collection of one-dimensional CW complexes, each made up of vertices, called switches, and edges, called branches, disjointly imbedded in F. Branches are open one-cells by convention so that r is the disjoint union of its switches and branches. If 6 is a branch of r and p £ b, then by a half-branch of 6, we mean a component of b — {p}. Two half­ branches of a given branch b are regarded as equivalent if their intersection is again a half-branch of 6, and the equivalence class is called an end of b. Thus, there are two ends of each branch of r. An end e of a branch b is said to be incident on a switch v of r if v lies in the closure of a half-branch representing e. The valence of a switch of r is the number of distinct ends which are incident on it. We say that r C F is a train track (or simply a track) in F provided that the following conditions hold. (1) (Smoothness) r is C 1 away from its switches. For each switch v of r, there is furthermore a line Tv(r), called the tangent line to r at v in the tangent plane to F at u, so that the following condition holds for each half-branch b whose closure contains v: a C 1 parametrization /?: (0,1) b with lim ^ i /3(t) = v extends to a C 1 map on (0 , 1] so that the one-sided derivative at 1 lies in T v ( t ). We may take the parametrization /? in such a way that w = limt_ i /3'(t) ^ 0. The unit vector in the direction of w lies in Tt,(r), is independent of the choices above and is called the one­ sided tangent vector to the end e corresponding to b. (2) (Non-degeneracy) For any switch v of r, there is an imbedding / : ( 0 , 1) —►r with / ( | ) = v which is a C 1 map into F. Thus, no 4

5

TRAIN TRACKS

switch of r is univalent, and we moreover demand that each simple closed curve component of r contains a unique bivalent switch and that every other switch of r is at least trivalent. (3) (Geometry) Suppose that S' is a component of F — r, and let D(S) denote the double of S along the C l frontier edges of S. Thus, non-smooth points in the frontier of S give rise to punctures of D(S). We require that the Euler characteristic x(D (S)) of D(S) be negative. Examples of train tracks may be found in Figure 1.1.1.

FIG U R E 1.1.1 Condition (1) requires that r C F be a “branched one-submanifold” of F rather than just a CW complex imbedded in F. Condition (2) rules out “dead ends” along r and prohibits unnecessary switches while introducing the convenient convention of requiring one bivalent switch on each curve component of r. We will comment further on condition (3) below. Suppose that D is a closed disk imbedded in the plane whose boundary is piecewise C 1 with n > 0 discontinuities in the tangent. If i: D —►F is a C 1 immersion which is an imbedding of the interior D of D into F , then 0 the image i(D) is an imbedded n-gon in F. In case i is only an immersion on 0 ° # # 0 F ,th e n i{D) is an immersed n-gon in F. In either case, i(D —D ) is called the frontier of the n-gon. The image under i of a non-smooth point in the frontier of D is called a vertex of the n-gon, so the vertices canonically decompose the frontier of an n-gon into n smooth arcs, which are called 0 frontier edges of the n-gon. Similarly, if X C D is a finite set of cardinality 0 0 m and i : D —X —* F is as above, we analogously define the image i(D —X ) to be an immersed or imbedded m-times-punctured n-gon in F. We also de0 fine an immersed or imbedded n-gon-minus-a-disk to be the image i(D —X ) C F where X C D is an open disk whose frontier is C and lies in D ; in i

.

.

0

.

.

.

•

0

1

•

0

6

S E C T I O N 1.1

particular, a nuUgon-minus-a-disk is called an annulus in F . Condition (3) in the definition of train track is equivalent to the condi­ tion that no component of F —r is an imbedded nullgon, monogon, bigon, once-punctured nullgon, or annulus. In particular, a train track can have no curve component homotopic into a puncture, and, furthermore, no two (distinct) curve components of a train track can be homotopic. If cr and r are train tracks in F and a is contained in r as a point set, then we say that cr is a subtrack of r and write a C r. In this case, cr can be obtained from r by deleting certain branches, amalgamating any two (not necessarily distinct) branches which meet at a resulting bivalent vertex, and then inserting the bivalent switch on each curve component of cr. Define a (finite) trainpath on r to be any C 1 immersion p:[nym] —► F , where n < m and n, m E Z5, with image contained in r so that the restriction of p to each subinterval (fc, k + 1) C [n,m] with k E 2Z, is a branch of r with p(k) and p(k + 1) switches of r. The integer m — n is called the length of /?, and p is said to be closed provided v = p(m) = p(n) and the tangents p'(m) and //(n ) (both of which lie in the tangent line to r at r) are non-zero and point in the same direction. Analogously, in case n = —oo, m = -foo or both, we define semi-infinite and bi-infinite trainpaths, respectively. Two trainpaths p \\I \ r and p 2'. h t are regarded as equivalent if there is an orientation-preserving homeomorphism :lR —» IR (or perhaps the restriction of such a map) so that p\ = p2 o, and when no ambiguities could develop, we may also simply identify a trainpath with the image of an underlying function. The reverse of the trainpath t i—►p(t) is the trainpath t ' i—►p(—^'), where the parameter t' is restricted to lie in a range so that this definition makes sense. If p is a trainpath in the train track r C F } F is some cover of F , and t is the full pre-image of r in F , then we will also refer to a lift of p to F as a trainpath on f. Notice that if cr C r, then a trainpath on cr gives rise to a trainpath on r in the natural way. P ro p o s itio n 1 .1 .1 : Suppose that R is a surface whose boundary d R (if any) is piecewise C 1, r is a train track in the surface F, and i \ R —* F is 0 a C immersion on R so that the restriction of i to each smooth curve or arc in dR is a finite trainpath on r. Then x(D (R )) < 0, where D(R) is the double of R along the C 1 arcs and curves in dR. P roof: We can easily add branches to r to produce a train track f D r in F so that each component of F —t is either a trigon or a once-punctured monogon (where the number of components of the latter type is s), and it clearly suffices to prove the proposition for f. Construct a foliation of each

TRAIN TRACKS

7

complementary trigon T with one three-pronged singular point V G T so that the leaves of the foliation are transverse to the frontier edges of T as in Figure 1.1.2a; similarly, foliate each complementary punctured monogon M with one three-pronged singular point U G M as in Figure 1.1.2b. These foliations combine in the natural way to produce (after some smoothing) a foliation T of F (which is C 1 away from singular points such as (7, V above) so that T is transverse to f.

FIG U R E 1.1.2 The pull-back i*(F) of T to R is a foliation of R which has only three0 pronged singularities in R ] suppose that there are p > 0 such singularities and q > 0 discontinuities on OR. Consider the double cover R of R branched 0 over the singularities in R ] of course, by the Riemann-Hurwitz formula,

x(R) = 2X(R)-p. The foliation i*(IF) lifts to a foliation of R which is C 1 on the interior except for p six-pronged singularities; furthermore, there are 2 q discontinuities on dR. Let us double R along each of its boundary components (in the usual sense of doubling) to produce a surface R. The foliation of R , in turn, gives rise to a foliation of R , and we can perturb the latter to finally produce a foliation T of R which is C 1 except for 2p six-pronged singularities, say {V}}, and 2q centers, say {£/,*}, one center arising from each non-smooth point of dR. Since the number of singularities is even, it is easy to produce a vectorfield on R which is tangent to f with zero set {[/,•} U {Vj} so that each Ui has order one and each Vj has order -2. By the Poincare-Hopf Theorem X ( R ) = 2 q - 4p,

8

S E C T I O N 1.1

and so

x(R) = \x(R)

=

9 - 2 p.

Finally, since X(D(R)) = 2X(R) - q, we find X(D(R))

= x(£) + p - q = q - 2 p + p - q = - p < 0 ,

as was claimed.

q.e.d.

C o ro llary 1.1.2: I f r C F i s a train track, then there can be no immersed nullgon, monogon, bigon, once-punctured nullgon, or annulus in F each of whose frontier edges is a finite trainpath in r. Furthermore, if f is the full pre-image of r in the universal cover of F, then any trainpath on t is imbedded. P ro o f: For any of the surfaces R mentioned above, x(D (R )) is nonnegative, so the first assertion follows immediately from the previous proposition. For the final assertion, if p is a trainpath on t which is not simple, then a finite sub-trainpath of p forms the frontier of an imbedded monogon or nullgon in the universal cover; this region projects to an immersed monogon or nullgon in F , and this contradicts the result just proved. q.e.d. C o ro llary 1.1.3: A surface F = F* contains a train track if and only if x{F) < 0 (i.e., 2g s — 2 > Q) and F is not the surface Fq (i.e., if g = 0 , then s > 3). Furthermore, if r C F is a train track with v switches and b branches, then we have the inequalities v < —6x (F ) —2 s, b < - 9 X(F) - 3s. P roof: That the stated conditions are necessary for F to contain a train track follows from the previous proposition (where dF = 0, so x{D(F)) = 2x(F )) and the observation that Fq contains no train track (since every essential curve in this surface is puncture-parallel). Sufficiency is easily established by producing an essential non-puncture-parallel curve in F (and adding a bivalent vertex) to produce a train track in F. For the second part, we may complete r to a train track f as in the proof of Proposition 1.1.1 and then alter f slightly near its switches as in Figure 1.1.3 to produce a train track f each of whose switches is trivalent.

T R A IN T R A C K S

9

If f has v switches and 6 branches, then v > v and b > 6, and we let t denote the number of trigons complementary to f in F. Since each switch of f is trivalent, 26 = 3t>, and since there are s complementary punctured monogons, 3t + s = v. Meanwhile, it was proved above that t + s = -x(D(F)) = - 2 X(F), so the first inequality follows from v a if r carries cr, and notice that < is a transitive relation on train tracks. Furthermore, if a < r, then a closed trainpath on a gives rise to a closed trainpath on r in the natural way. Multiple curves carried by r may be described by labeling the branches of r with integers: for each branch 6, pick p E 6 and write pc(b) for the number of points in ~1(p) fl C, where is the supporting map of the carrying C < r. The definition is independent of the choice of p E 6, and these integers satisfy a condition defined presently. (In fact, p c is also independent of the choice of supporting map; see Proposition 1.7.5.) 10

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11

For each switch v of r, fix a direction in the tangent line Tv{r) to r at v. The end e of a branch 6 of r which is incident on v may then be called incoming if the direction of the one-sided tangent vector to e at v agrees with this direction, outgoing if not. The function fie defined above gives rise to a function defined on ends of branches by the rule /ic (e) = A*c(&) if e is an end of 6. If e i , . . . , er are the incoming ends of branches incident on v and er+ i , . . . , er+* are outgoing, then this function satisfies the following switch condition at v: i) + ... + fJ>c(er) = ^ c (e r+i) + ... + fJ.c(er+t)Insofar as fic satisfies the switch condition at each vertex of r, we say simply that fie satisfies the switch conditions (on r). Conversely, any assignment /i of a nonnegative integer to each branch of r which satisfies the switch conditions defines a multiple curve as follows. Choose a regular neighborhood TV of r in F and arrange /i(6) arcs parallel to the branch b disjointly imbedded in TV as in Figure 1.2.1a. If v is a switch of r, the number of endpoints of arcs so constructed corresponding to incoming ends of branches agrees with the number corresponding to outgoing ends by the switch condition at v. There is thus a unique way to combine these arcs in N near v, connecting incoming to outgoing, so that the result is a collection of arcs disjointly imbedded in TV; see Figure 1.2.1b. Performing this construction near each switch of r yields a collection of simple closed curves disjointly imbedded in N C F. Since there can be no immersed nullgon or punctured nullgon in F whose frontier consists of a finite trainpath on r by Corollary 1.1.2, it follows that no curve in the collection just constructed is null-homotopic or puncture-parallel; thus, the collection is in fact a multiple curve, as was asserted.

FIG U R E 1.2.1 We must generalize the material introduced above for later application as follows. Suppose that C C F is a collection of closed curves immersed in F; components of C need not be simple and may intersect one another in F. If r C F is a train track, then we say that r carries C if there is a

12

S E C T I O N 1.2

homotopy t, for 0 < t < 1, with o the given immersion of C in F, so that = \ satisfies conditions (1) and (3) above; is called the supporting map of the carrying. As in the previous paragraph, if C is carried by r, then no component of C can be null-homotopic or homotopic into a puncture by Corollary 1.1.2. Furthermore, if C is carried by r, then there is an induced [ZZ+ U {0}]-valued function pic defined as before, where the count of points in ~~l {p) fl C (for p a regular value of the supporting map ) must be performed with multiplicity. It is clear that such a function p c must satisfy the switch condition at each switch of r, and by the previous paragraph, therefore gives rise to a multiple curve, say C ' C F. In particular, a closed trainpath p on r gives rise to a (connected) curve, say C, immersed in F which is carried by r, and C in turn gives rise to a multiple curve C' (which need not be connected), as above. Notice that if p traverses a branch b of r exactly times, then pc(b) = p c (b ) = Define a multicurve in F to be the isotopy class of a multiple curve in F, and let S (F ) denote the set of all multicurves in F. We say that a multicurve is connected if some underlying multiple curve is connected; the collection of connected multicurves in F is denoted S '(F ) C $(F ). The geometric intersection number extends to a function i:S (F )xS (F )-+ 7 Z + U {0 } in the natural way. We say that a train track r carries a multicurve if it carries some representative multiple curve. (There is actually a one-to-one correspondence between the collection of all multicurves carried by a train track and the collection of all [Z5+ U {0}]-valued functions which satisfy the switch conditions defined on the set of branches of r; see Theorem 1.7.12.) In a 1922 Breslau lecture [D], Dehn described a one-to-one correspon­ dence between the set S(F*) of multicurves in a surface of negative Euler characteristic and a subset of TLM, for M = 6g — 6 + 2 s; we call this result Dehn’s Theorem. In 1976, Thurston rediscovered Dehn’s result and extended it to a parametrization of (Whitehead equivalence classes of) mea­ sured foliations in F. (See [FLP] for the definitions of measured foliation and Whitehead equivalence as well as a version of Thurston’s coordinates.) We will call this more general result the Dehn-Thurston Theorem and will derive the analogue of this theorem in the setting of train tracks in Theo­ rem 3.1.1. The remainder of this section is devoted to a discussion of Dehn’s Theorem itself; sufficient detail is given that the reader could (and should) supply a complete proof. [FLP] contains a proof of the Dehn-Thurston Theorem in the setting of measured foliations (this theorem follows from our train track version), and [PI] introduced the version of Dehn’s Theorem which we discuss here. In the Addendum, we briefly describe the compu­ tations of [PI], which give the natural action of the mapping class group of

M U L T I P L E C U R V E S A N D D E H N ’S T H E O R E M

13

F (that is, the isotopy classes of orientation-preserving homeomorphisms of F) on Dehn’s coordinates for multicurves in F. A pair of pants is a compact planar surface of Euler characteristic -1; thus, a pair of pants is homeomorphic to a closed disk minus two open disks whose closures are disjoint and lie in the interior of the closed disk. Choose an oriented “standard” pair of pants P which is C 1 and has C 1 boundary. Label the boundary components d{ and choose closed arcs C Si, called windows, for i = 1,2, 3 as in Figure 1.2.2a.

FIG U R E 1.2.2 Let D be a collection of arcs properly imbedded in P with dD contained in the interior of the windows so that no component of D is boundaryparallel (i.e., properly homotopic into d P ). We say that two such collections are parallel if they are related by a proper isotopy of the identity map P —►P which fixes each point of d P —U{wi, w2, w3}. A basic fact.(left as an exercise) due to Dehn is that there is a proper isotopy of D (not necessarily respecting the windows) which takes D into a disjoint union

14

S E C T I O N 1.2

of parallel copies of the arcs illustrated and labeled in Figure 1.2.2. The different possibilities are uniquely determined by the number m,- of times that D intersects i = 1,2,3, subject only to the restriction that the sum mi + m2 + m3 is even. Moreover, D is parallel to a disjointly imbedded collection of arcs so that each arc in the collection is parallel to an arc obtained from one of those illustrated in Figure 1.2.2 by performing Dehn twists along certain of the components of dP. For each triple (m i, m2 , m3 ) with mi -f m2 -h m3 even, let us choose a collection of imbedded arcs (with no twisting and with endpoints in the interiors of the windows), which represents the corresponding proper isotopy class of one-submanifold of P. The simple structure of one-manifolds properly imbedded in P suggests that we decompose an arbitrary surface F = F* (of negative Euler charac­ teristic) into pairs of pants. A pants decomposition {A,} of F is a multiple curve in F so that each component R of A —U{A,-} is homeomorphic to the interior of a pair of pants. We do not require the closure of R in F to be an imbedded pair of pants. Some examples of pants decompositions are given in Figure 1.2.3. Each component A* of a pants decomposition is called a pants curve. By considering the Euler characteristic, one finds that there are N = 3g —3 + s pants curves in a pants decomposition of F = F* and M = 2g —2 -f s complementary regions. In the following discussion, the subscript i ranges from 1 to A, and the subscript j ranges from 1 to M .

F IG U R E 1.2.3 For each pants curve A,*, choose a small closed regular neighborhood Ai of K{. Let A = S1 x [—1 , 1] denote the “standard” oriented annulus and let G be the canonical projection collapsing A onto the core S1 x 0. Choose for each index i an orientation-preserving homeomorphism of A onto A{ which maps S1 x 0 to A'*; V{ is called the characteristic map of the annulus A(. In each A,-, choose a closed arc a*. Choose an open regular neighborhood U of the set A of punctures of F so that the frontier oiU consists of smooth curves, one about each puncture. 0 Each component Pj of F —U — U{ A i] is a pair of pants imbedded in A, and we may choose for each Pj an orientation-preserving homeomorphism fj of P onto Pj which carries each component of f ~ x 01/i o G ' 1 to

M U L T I P L E C U R V E S A N D D E H N ’S T H E O R E M

15

a window in 5 P , whenever Ai fl Pj ^ 0. The homeomorphism fj is called the characteristic map of Pj. Now, let C be a multiple curve in F representing some multicurve, and for each index i, let m, = i(C,Ki), where we assume that C and K{ hit efficiently in the sense that there is no imbedded bigon in F whose frontier consists of one arc from each of C and Ki\ in this case, one observes that C and / 0, for each i — 1 ,..., N . (b)If K ix, Ki2, are pants curves which together bound a pair of pants imbedded in F*, then the sum mtl + m ;2 + m ;3 of correspond­ ing intersection numbers is even. Furthermore, the intersection number corresponding to a pants curve which bounds a torus-minusa-disk or twice-punctured nullgon imbedded in F'* iseven. As we have seen, to parametrize multicurves, we must make a large number of conventions. We define a basis A for multicurves on F to be a choice of pants decomposition of F , choices of associated characteristic maps, and a choice of model arcs on P as in Figure 1.2.2. In practice, to specify a basis, we fix a choice of model arcs in P once and for all, regard F as imbedded in IR3, and draw pictures. The characteristic map Vi is the trivialization of the normal bundle to K{ C F C IR3 that extends across a disk in IR3 with boundary K{. We draw and label K{ and and draw f j i f 12) and fj(£ 13), labeled 2 and 3 respectively, in each Pj. E xam ple: Consider the basis for ^(F g) indicated in Figure 1.2.4a. We will draw a representative multiple curve C of the multicurve with Dehn parameter values (m l5 m 2, m3 ) x ( t i , t 2) £3) = (5,1,2) x (1, —1, 0). There are five components of C fl A\ since m\ = 5, and one of these twists to the right since t\ = -fl. Similarly, there is one component of C fl A 2 since m 2 = -f 1, and it twists once to the left since £2 = —1 ; there are two components of C fl A 3 since m3 = 2 and no twisting since £3 = 0. Thus, we draw our representative C in each of the annuli A{,i = 1,2,3, as in Figure 1.2.4b. We then connect up these arcs uniquely using the images under f \ of arcs parallel to ^13, £23>^33 and under f i of arcs parallel to ^12, ^23,^22 as shown in Figure 1.2.4c.

(a)

(b) F IG U R E 1.2.4

(c)

M U L T I P L E C U R V E S A N D D E H N ’S T H E O R E M

17

For later use, we define a standard basis A sg on F* in Figure 1.2.5 (using the arcs in Figure 1.2.2).

s = 0

s > 1 FIG U R E 1.2.5

§1.3 R E C U R R E N C E A N D T R A N S V E R S E R E C U R R E N C E

The train track r C F is called recurrent if for each branch b of r, there is a multiple curve Cb carried by r with supporting map :F —» F so that 6 C 0. Some examples of recurrent and non-recurrent train tracks are given in Figure 1.3.1. (The verifications are routine and left as exercises.) The choice of terminology comes from the fact that the bi-infinite trainpath on r associated with Cb passes through 6 infinitely often, so b “recurs” along this path. Conversely, suppose that p: (—oo, oo) —* r is such a trainpath so that p = p(r) = p(t) E b and the tangents p'(r) and p'{t) have the same direction at p for some r < t. Thus, p is a closed trainpath on r which traverses 6, and the corresponding multiple curve Cb (see §1.2) satisfies the conditions above. Furthermore, it is not difficult to check that r is recurrent if and only if it carries a multiple curve C with pc{V) > 0 simultaneously for every branch 6: indeed, if for each branch 6, Cb is a multiple curve so that p c b(b) > 0, then the sum p = Ylb V’Cb satisfies the switch condition at each vertex (since the switch condition is linear), and p determines an appropriate multiple curve C. Notice that if r is an arbitrary train track, then there is a canonically defined maximal recurrent subtrack a C r: a branch b of r lies in a if and only if there is a multiple curve C in F with pc(b) > 0.

non-recurrent

recurrent

FIG U R E 1.3.1 A transverse measure (or simply a measure) p on r is a function which assigns to each branch 6 of r a nonnegative real number p(b) E IR+ U {0} 18

RECURRENCE AND TRANSVERSE RECURRENCE

19

which satisfies for each switch v of r the switch condition fi(ei) + . . . + fi(er) = /i(er+i) + . . . + n(er+t),

where e i , ... er are the incoming ends of branches which are incident on v and er + i , ... er+* are the outgoing ones (and the measure of an end is the measure of the corresponding branch, as before). The pair (r, p) is called a measured train track. If 6 is a branch of r, then the number p(b) is called the weight on (or measure of) 6, and we write simply p > 0 if each branch has nonzero weight. A measure which is [Z+ U {0}]-valued is called an integral (transverse) measure. An integral measure on a train track describes a multicurve in F (as in §1.2). More generally, when all the weights are rational, a measured train track describes a weighted multicurve, as we shall see. Still more generally, if the weights are not rationally related, then a measured train track describes a “measured geodesic lamination” (to be defined in §1.7). Let n denote the number of branches of r. Of course, the set of all fitvalued functions defined on the set of branches of r is naturally identified with fitn, and the collection of such functions which satisfy the switch con­ ditions is a sub-vector space H C fitn since each switch condition is linear. The collection of all transverse measures on r is then # fl[IR+ U {0}]n, and the collection of all measures which are strictly positive is V = H f lf itj. Now, if r is recurrent, then it supports a measure p > 0 as we saw above. Conversely, if there is a measure p > 0 on r, then V is not empty and is therefore homeomorphic to an open cell of the same dimension as H itself. Inside V , we may then find a point with rational coordinates; multiplying all weights to clear denominators thus gives an integral point in V , and this point describes a multiple curve C C F with p c > 0, so r is recurrent. Thus we have proven P ro p o s itio n 1.3.1: A train track r is recurrent if and only if it supports a transverse measure p > 0 . Suppose that a is either another train track in F or perhaps a mul­ tiple curve in F which intersects r transversely (if at all). We say that a hits r efficiently provided that no component of F — cr — r is an imbed­ ded bigon in F. A seemingly stronger characterization of hitting efficiently (which is quite useful in the sequel) is provided by P ro p o s itio n 1.3.2: Suppose that r C F is a train track and a C F is either another train track or perhaps a multiple curve which meets r transversely (if at all), and let a and f, respectively, denote the full pre-images of a and r in the universal cover F of F. Then a hits r efficiently if and only if there is no imbedded bigon in F whose frontier is contained in a U r.

20

S E C T I O N 1.3

P ro o f: It is obvious that the stated condition is sufficient for a and r to hit efficiently. Conversely, suppose B C F is such a bigon. By the first part of Corollary 1.1.2, it must be that one frontier edge, say E a, of B lies in a and the other, say E Ti in f. Any trainpath p on f is imbedded in F by the second part of Corollary 1.1.2. Furthermore, since B has compact closure (by definition of an imbedded bigon) and r C F is closed, the closure of each component of p fl B must meet the frontier of B in two distinct points. Any such trainpath which meets both E r and B must also meet E a, for otherwise there would be an imbedded monogon or bigon in F whose frontier edges consist of finite trainpaths on f, in contradiction to the first part of Corollary 1.1.2. Thus, any trainpath on f which meets B gives rise to an imbedded sub-bigon of B which has one frontier edge in a and one frontier edge in f. The analogous assertion also holds for 3, and we proceed by induction on m. First, suppose that m = 3 and let £i,£2 ,£3 , respectively be the total tangential measures corresponding to each of the frontier edges of R. The equations 3 £* = ^ ^ nij , f°r 2 =

1 )2 , 3 ,

have a unique nonnegative solution since the £,• are constrained by Condi­ tion (1) in the definition of tangential measure; namely nij =

+ f j - 6 ) for

= {1,2,3},

and this solution is integral by the evenness condition. The basis step is completed by connecting the ith and j th frontier edges of R with arcs as in Figure 1.3.4b, for distinct i, j £ {1,2,3}. For the inductive step, suppose that R is an m-gon with m > 3, and let £1, . . . ,£m, respectively, denote the total tangential measures of the frontier edges of R enumerated in the order in which they occur in a C° traversal of the frontier of R ; let us also regard the indices as determined mod m, so that, for instance, £m+i = £i- Select i £ { 1 ,..., m} so that £,• +£*+i is a minimum of {£;- + £ j+ i : j = 1 , . . . , m};

26

S E C T I O N 1.3

and add a branch b to r " separating the ith and (i + l) st frontier edges from the rest as in Figure 1.3.4c, so that b decomposes R into a trigon and an (m —l)-gon. Select £* + points on the new branch 6, and complete in each of these regions using the inductive hypothesis to construct the required collection of arcs in ii!. By construction, the family of arcs produced above combine to give a collection C of simple closed curves in F , and we can easily arrange that C is C 1. Furthermore, C meets each branch 6 of r" in exactly i/ff(b) points, each component of C intersects r " transversely in a non-empty set, and no component of F — (C U r h ) is a bigon. To complete the proof of the lemma, we must show that C is a multiple curve (i.e., no component of C is null-homotopic or puncture-parallel). If some component c of C is nullhomotopic, then c must bound a disk in F; since r " meets c transversely, r" must intersect the interior of this disk, and we find an imbedded bigon in F so that one frontier edge lies in r" and the other in c. As in the proof of Proposition 1.3.2, there is an innermost such bigon, which must be disjoint from C U r " and is therefore a component of F —(C U r " ) , which is impossible. In the same way, no component of C can be puncture-parallel, so C is in fact a multiple curve, and the lemma is proved. q.e.d. R em ark s: 1) If each complementary region of r in F is either a trigon or a once-punctured monogon and v is an even tangential measure on r, then there are essentially no choices in the construction above of a multiple curve hitting r efficiently and realizing i/(b) = card (C fl b) for each branch b of r; the underlying multicurve is actually unique, as we shall see later. However, if some component of F — r is other than above, then this is not true. Furthermore, in contrast to transverse measure, different tangential measures on the same track can give rise to the same multicurve. These points will be addressed in §3.4. 2) The last step in the argument above proves the general fact that if a collection C of simple curves disjointly imbedded in F hits a train track in F efficiently, then C is in fact a multiple curve. An analogue of Proposition 1.2.2 for transversely recurrent tracks (see also Theorem 1.4.3) is contained in the following C o ro llary 1.3.5: Suppose that r C F is a train track. conditions are equivalent: (i) r is transversely recurrent.

The following

RECURRENCE AND TRANSVERSE RECURRENCE

27

(ii) r supports an even (integral) tangential measure v > 0 . (in) There is a multiple curve C C F hitting r efficiently so that C f\b ^ 0, for every branch b of r. P roof: By definition of transverse recurrence, we can find for each branch b of r a multiple curve Cb C F so that Cb hits r efficiently and b fl Cb ^ 0The multiple curve Cb determines an even tangential measure Vb = vch as before. The sum v =

£

**

b a branch of r

is a tangential measure on r (by linearity of the inequalities in the definition of tangential measure) which is even, so (i) implies (m). Furthermore, the multiple curve C C F which arises by applying the previous lemma to v meets each branch b of r since i/(b) / 0, so (ii) implies (in'). Since (Hi) obviously implies (i), the proof is complete. q.e.d. E xam ple: Armed with Corollary 1.3.5, we now verify that the train track r C F = F l in Figure 1.3.2a is not transversely recurrent. Adopt the notation in the figure for branches of r, so that one component of F — r is a once-punctured monogon and the other is a trigon, whose three frontier edges correspond to finite trainpaths on r which traverse branches d/, edbabcfy and ec, respectively. If r were transversely recurrent, then by the previous result, it would support an even tangential measure v > 0 . By Condition (1) in the definition of tangential measure, we must have i/(e) + v(d) + 2 1/(6) + v(a) + i/(c) + i/(f) < v(c) + i/(e) + i/(d) + i/(/), so v(a) = v(b) = 0, contradicting that v > 0. This establishes that r is not transversely recurrent. The verification that the track in Figure 1.3.2b is not transversely recurrent is similar and left as an exercise. We say that a train track is maximal if it is not a proper subtrack of any other train track. Of course, we can add branches to any track which is not maximal until each complementary region is either a trigon or a once-punctured monogon (as in the proof of Proposition 1.1.1). We say that a birecurrent train track is complete if it is not a proper subtrack of any birecurrent track, and a more interesting problem involves extending a birecurrent train track to a complete train track. T h e o re m 1.3.6: (a) I f g > 1 or s > 1, then any birecurrent train track on F = Fg is a subtrack of a complete train track, each of whose complemen­ tary regions is either a trigon or a once-punctured monogon.

28

S E C T I O N 1.3

(b) Any birecurrent train track on F = is a subtrack of a complete train track, whose unique complementary region is a once-punctured bigon. P roof: To begin the construction of a complete birecurrent train track with a given birecurrent train track r as subtrack, we can add curve components to r as in the first step of the proof of Lemma 1.3.4 to produce a train track r 7 so that each component of F — r 7 is an m-gon, a once-punctured m-gon, an m-gon-minus-a-disk, or a pseudo pair of pants. Recurrence of r 7 follows immediately from the recurrence of r, and transverse recurrence of r 7 is proved as follows. By Corollary 1.3.5, we may choose a strictly positive even tangential measure on r which we may multiply by two to produce an even tangential measure v so that v(b) > 2 for each branch b of r. Let us now apply the inductive argument given at the beginning of the proof of Lemma 1.3.4 with a small difference: wherever we before extended v by taking a tangential measure zero on an added curve, we extend this time by taking tangential measure 2. As before, this produces a strictly positive tangential measure on r 7 which satisfies the hypotheses of Lemma 1.3.4, and application of this lemma finally proves that r 7 is transversely recurrent, as was claimed. We next proceed to add branches across complementary regions of r 7, and some care is needed to guarantee that the result is birecurrent. We will add branches to r 7 in a complementary region R in one of the following ways: M ove 1: If R is an m-gon, m > 3, then add a branch in R con­ necting two vertices which are not consecutive along the frontier; see Figure 1.3.5(1). M ove 2: If R is a once-punctured m-gon, m > 1, then add a branch in R encircling the puncture with both endpoints at a sin­ gle vertex in the non-smooth component of the frontier of R\ see Figure 1.3.5(2). M ove 3: If R is an m-gon-minus-a-disk, m > 1, then add a branch in R which connects two switches of r, one in each of the two components of its frontier; see Figure 1.3.5(3). M ove 4: If R is a pseudo pair of pants, then add a branch in R connecting any two switches of r lying in distinct components of the frontier of R\ see Figure 1.3.5(4). M ove 5: If R is a pseudo pair of pants, then add a branch e in

29

RECURRENCE AND TRANSVERSE RECURRENCE

R running from a switch of r in the frontier of R back to itself so that each component R —e is either a once-punctured monogon or a monogon-minus-a-disk; see Figure 1.3.5(5). M ove 6: If R is a once-punctured m-gon, m > 2, and 6 is a frontier edge of R with vertices v and w, then add a branch e in R connecting v to w so that the component of R —e containing the puncture of R has e and 6 as its frontier edges; see Figure 1.3.5(6).

( 2)

( 1)

e tc .

or (3)

or

e tc .

(4 )

( 6)

(5 ) FIG U R E 1.3.5

It is obvious that the application of any of these moves to a train track produces a train track with exactly one new branch, and we furthermore

30

S E C T I O N 1.3

claim that these moves preserve transverse recurrence. It certainly suffices to consider the case that a train track 1) and p |(n_ i>n), respectively, agree with the prescribed orientations on e and e'. P r o o f: To prove the first basic fact, fix the branch e and let E denote the closure o f the union o f branches e; adm itting such a train path starting from e. Suppose that 61 C E contains a sw itch v o f a in its closure and the spec-

32

S E C T I O N 1.3

ified orientation on 61 corresponds to the incoming (outgoing, respectively) direction at v\ if 62 is another branch of a so that the closure of 62 contains v and the specified orientation on 62 corresponds to the outgoing (incoming, respectively) direction at v, then we must have 62 C E as well. Suppose, to reach a contradiction, that there is some branch b C r — E whose closure contains some point of E. By recurrence, there is some connected multiple curve C carried by cr with fic(b) ^ 0 , and C inherits an orientation from the orientation of -k + Ai,

(ii) I f the rji are varied by an amount Arji such that 0 < 77^-h A.77z- < then A C = Y a O(Arji), (in) a = H \ 0 (e~Lr]i), and / 3 = H i 0 (e~Lr]i), (iv) A a =

0 {e~ LAi)i), and A/? = H i 0 (e~ LAf]i),

where the various constants may depend on k. P roof: We proceed by induction on k. Because v > 0 is an integral tan­ gential measure, v(b) > 1, for each branch b, so making L large will make all the branches long. In the case k — 1, the existence of the triangle is im­ mediate, and the estimates are simply those of Lemma 1.4.5. When k > 1 ,

GENERICITY AND TRANSVERSE RECURRENCE

51

we break the sliver up into a sliver with a smaller number of sides and a triangle as in Figure 1.4.10. The exterior angle 77of the triangle is simply rjk + c*o- If L is sufficiently large, then induction hypotheses (in) and (iv) imply that ao < f and a 0 + Ac*o < so that 77< | and 77+ A 77< | , and hence Lemma 1.4.5 applies to the triangle.

FIGURE 1.4.10

As to the estimates:

(i) follows immediately from Lemma 1.4.5.i and induction hypothesis (i).

For (ii), A 77 = A 77J; -I- Ac*o = Yli O(Arji) by induction hypothesis (it;); so by Lemma 1.4.5.ii, A C = C(At7) + O(AA), and estimate (if) follows immediately from these formulas and induction hypothesis (ii).

For (m ), we have 77 = 77^ -f cto = ]Ci OiVi) by induction hypothesis (Hi)] so by Lemma 1.4.5.iii, we have a = 0 ( e “ L77) = J2iO (e~Lrji). The estimate on /? follows by symmetry.

For (it;), we have Act = 0 ( e ~ LArj) + 0 (e~ L A A ) by Lemma 1.4.5.iv. Using the formula above for A 77 and induction hypothesis (ii), the estimate on A a follows. Again the estimate on A/3 follows by symmetry. q.e.d.

52

S E C T I O N 1.4

FIGURE 1.4.11 L e m m a 1 .4 .7 : C on sider a hyperbolic triangle with edge-lengths and angles as indicated in Figure I f A, B , C , and A + B — C are sufficiently large, then we have the estim ates /• \

^

(i) y ~ 2 e

C-A-B 2

(ii) I f A ,B , and C are varied by bounded amounts A A ,A B , and A C , respectively, then Ay ~ O ( e ^ A C ) +

A A) + 0 ( e ^ A 5 )

P ro o f: For (i), the fundamental estimate (*) in the proof of Lemma 1.4.5 says C = A + B + log

+ o (l)

= A + B + logsin 2 ^ + o (l), SO

y

sin-~e and so

c-a-b

.

C-A-B

7

2~6

2 5

as desired. For (ii), the hyperbolic law of cosines gives cosh A cosh B — cosh C CO ST —

7

so

-------------------------------------------------------

sinh A sinh B

dy sinh C —sin 7 •— = — : dC sinh A sinh B

GENERICITY AND TRANSVERSE RECURRENCE

and hence

dy

— Q yo,

ec - A - B ^

C/0

C-A -B —

C-A -B —

£

53

2

p C

2

Similarly, dy = cosh B cosh A : . sinh A, {coshf A .cosh . . B_ — cosh, C^ } , —sin — o A sinh B sinh B _ 2

7

hence

dy dA

— l

^

pC-A-B

— p

c—A— B 2

The symmetric formula follows for Since A, B y and C vary by bounded amounts, = 0{e~Si a ) in the perm itted range. The same holds for the other partials, and the result follows from Taylor’s theorem. q.e.d. 2

F IG U R E 1.4.12 L em m a 1.4.8: Suppose that angles 0 < ,• < ^ are given. If L is suffi­ ciently large, then there is a scalloped region R with these exterior angles (as in Figure l.f.12), where the length of a branch b is Lv(b). a,/?, and y are continuous functions of the rji, and 77

(i) a = (e“^ ) , 0

(it) A a =

^ *), 77

where identical estimates hold for f3 and y.

54

S E C T I O N 1.4

P ro o f: By Lemma 1.4.6, we can construct three slivers with long sides of respective lengths A ',# ', and C", and we let v(A)> v (B ) y and ^(C), respectively, denote the total v-tangential measures of the corresponding frontier edges of the associated complementary sliver. From Lemma 1.4.6.i, Lv(C) > C ',A f > Lv(A) — constant, and B ' > Lv(B) —constant, so A 1 + B f —Cf > L[v(A) + v(B) — v{C)\ — constant > L — constant, and similarly for the cyclic permutations of A', J9',C". Thus, if L is suffi­ ciently large, the triangle inequalities hold, and we can construct a hyper­ bolic triangle R f with side-lengths and opposite interior angles a ',/ 2' , 7 ', respectively. Furthermore, for L sufficiently large, Lemma 1.4.7 will apply to this triangle. The angle a is simply a f plus the angles at the extreme vertices of two of the slivers. By Lemma 1.4.6.iii and Lemma 1.4.7.i, these are both 0 (e~ s'), and estimate (i) follows. Thus, and hence by Lemma 1.4.7.ii A y ' = 0 ( e ~ * A A ') + 0(e~ * A S ') + 0 (c ‘ = A C') = J 2 0 (e-^Ar)i) by Lemmas 1.4.6 and 1.4.7. On the other hand, A7 is the sum of A 7 ; and the changes in the extreme angles of the slivers, which are at most Y 20(e~LAr]i) = ^ 0 ( e ~ s’A ^ ). Similar estimates hold for a and /?, and this completes the proof. q.e.d. P ro o f o f T h e o re m 1.4.4 (conclusion): Choose some application of Lemma 1.4.8 gives

L > Lq so

that

$i < min(£o, | ) , |A0,|
0 and L q > 0, we claim that there is a hyperbolic structure on F and a piecewise geodesic CW complex T C F so that the following conditions hold. -T is isomorphic to the underlying CW complex of r. -At each vertex of T, the angle of deviation from a straight line is less than Sq.

56

S E C T I O N 1.4

-Each edge of T has length exceeding

Lq.

Indeed, fix some e > 0 and L > 0, and consider the hyperbolic structure on F guaranteed by Condition (Hi). For each branch of r, choose some bi-infinite trainpath traversing it at least once; this trainpath lifts to a biinifinite geodesic curve in hyperbolic space with uniformly small geodesic curvature. As such, this bi-infinite curve lies uniformly close to a unique geodesic. Since r has only finitely many branches, r may be “unsmoothed” to a CW complex T C F, and given any £0 > 0 and L q > 0, £ and L can be chosen so that T satisfies the second and third conditions above. For each branch b of r, let i/(b) E IR+ denote the hyperbolic length of the corresponding edge of T. We claim that for £o sufficiently small and L q sufficiently large, v is actually a tangential measure on r as in Condition (ii). Indeed, suppose first that Q is a trigon component of F — r, and let R be the correspond­ ing component of F — T. By the second condition above, R has exactly three vertices with small interior angles (and the remaining interior an­ gles, if any, are near 7r), and we consider the triangle R' C R spanned by these vertices, where the sides of R f have respective lengths A', see Figure 1.4.12. Furthermore, let A, R, C, respectively, denote the sums of v-values of branches in the corresponding frontier edges of Q. Taking L q sufficiently large, Lemma 1.4.6.i applies to each component of R — R f. If £0 is sufficiently small, then the interior angles of R f are small, so we may arrange that A! + B f —C' exceeds 2n (by the fundamental estimate (*) in the proof of Lemma 1.4.5), where n is the number of branches of r. Thus, A + B > A' + B' > C + 2 n > C, and similarly for the other triangle inequalities, as desired. The proof that v satisfies the appropriate strict inequalities on a complementary m-gon, for m > 3 , follows easily by induction. Finally, if Q is an m-gon-minus-a-disk component of F —r, then we may first decompose Q into an (m -|-2 )-gon and a monogon-minus-a-disk (where each resulting component has interior angles near 0 and ir) by cutting along a geodesic arc in Q with both its endpoints at a vertex on the non-smooth component of the frontier of Q. One establishes the required inequality by arguing separately, as above, in each component. This completes the proof that Condition (Hi) implies Condition (ii) and hence also the proof of Theorem 1.4.3. q.e.d. R e m a rk : Notice first that the proofs above that Condition (in) implies Condition (ii) and Condition (ii) implies Condition (i) did not require the hypothesis of genericity on the train track. Genericity of the train track is

GENERICITY AND TRANSVERSE RECURRENCE

57

needed however to show that Condition (i) implies Condition (ii). Indeed, consider the non-generic train track r Q F§ illustrated in Figure 1.4.3 (and discussed in the example before Corollary 1.4.2). We saw that r is transversely recurrent, but does not support a tangential measure satisfying strict inequalities in Conditions (1) and (2) in the definition of tangential measure. This also shows that Condition (i) does not imply Condition (iii) without the hypothesis of genericity, as genericity was not needed in the proof that (iii) implies (ii). To close this section, we describe yet another method of producing a complete generic train track from a birecurrent generic one. Suppose that r C F is a (not necessarily birecurrent or generic) train track, R is a component of F —r, and a is an arc imbedded in R so that da is contained in the frontier of R and da is disjoint from the switches of r. Provided that no component of R — a is a bigon or trigon, we say that the arc a is admissible; see Figure 1.4.14a for an example. One can produce a new track r ' by collapsing a neighborhood of da in the frontier of R to a single arc, as indicated in Figure 1.4.14b; this modification is called a trivial collapse along a. Observe that if r is generic, then the collapse may be performed in such a way that r ' is generic as well.

FIGURE 1.4.14 We next show that birecurrence is preserved under such a trivial col­ lapse. Adopt the notation indicated in Figure 1.4.14 for the branches in the frontier of R . To any branch f of r other than a or d, there is a corre­ sponding branch / ' of r '. If /i is a (positive) measure on r, then there is a corresponding (positive) measure / / on r ' defined by r //(a), if / ' = a' or 6'; /i(d), if / ' = c' or df; H(a) + fi(d), if / ' = e'; , //(/), otherwise. It follows easily from Proposition 1.3.1 that a trivial collapse along an ad­ missible arc preserves recurrence. Moreover, suppose that r is transversely

58

S E C T I O N 1.4

recurrent and C is a multiple curve hitting r efficiently. If some sub-arc a of (some component of) C is admissible, then one sees easily (using parallel copies of components of C) that a trivial collapse of r along a preserves transverse recurrence, as was asserted. P ro p o s itio n 1.4.9: I f r C F is a birecurrent generic train track, then there is a complete generic train track r' > r so that r f arises from r by a composition of trivial collapses along admissible arcs. P roof: We claim that unless r is complete, there is an admissible sub-arc a of a multiple curve which hits r efficiently. Performing a trivial collapse along a produces a birecurrent track carrying r by the remarks above and forms the inductive step in an obvious argument; it therefore remains only to prove the claim. To this end, since r is transversely recurrent, there is some multiple curve C C F hitting r efficiently which meets each branch of r by Corol­ lary 1.3.5. Moreover, since r is generic, we may sneak up various compo­ nents of C to furthermore arrange that the following condition holds: if v is a switch of r giving rise to a non-smooth point in the frontier of a compo­ nent R of F —r, then there is an arc among the (closures of) components of C — r and a trigon component of F — (r U C) whose frontier contains both v and this arc. Now, suppose that R is a component of F — r so that x(D (R )) < —2. If no component of C — r in R is admissible, then it is an easy m atter to surger C along an arc in R to produce another multiple curve meeting r efficiently which does contain an admissible sub-arc in R\ see Figure 1.4.15. Thus, provided x(D (R)) < “ 2, we may find a suitable admissible arc in R.

F IG U R E 1.4.15

GENERICITY AND TRANSVERSE RECURRENCE

59

It remains to consider the cases in which a component R of F —r is a once-punctured bigon, a fourgon, or a twice-punctured nullgon, and in this last case, every arc among components of C — r is admissible. If R is a complementary fourgon, then consider the respective total i/c-tangential measures & , i = 1 ,..., 4, of the frontier edges of R enumerated in the order in which they occur in a C° traversal of the frontier, where vc is the even tangential measure on r associated to C. It is an exercise to check that provided £i -F £3 ^ £2 + £4 , there must be an admissible arc among the components of C — r. If equality holds in the previous equation, then choose a branch b' of r which occurs exactly once in the frontier of iZ, and define a newtangential measure v on r by v(b) = / Vc^ + £ if 6 = b'} I vc(P)> otherwise, where e is chosen sufficiently small that v is actually a tangential measure; such a choice of e is possible since vc satisfies strict inequalities in Condi­ tions ( 1) and (2 ) in the definition of tangential measure by construction. As before, we may take a rational approximation to 1/, clear denominators, and multiply by two to finally produce an even tangential measure on r so that the corresponding multiple curve contains an admissible sub-arc in R. Finally, consider the case of a complementary once-punctured bigon iZ, and let £ and £', respectively, denote the total i/^-tangential measures of the frontier edges of R. If £ ^ then C contains an admissible sub­ arc in R\ if equality holds in the previous equation and F ^ F}, then a modification as in the previous paragraph produces a suitable admissible arc in R. Thus, unless r is complete, we can find an admissible sub-arc of a multiple curve meeting r efficiently, so the claim and hence the proposition are proved. q.e.d.

§1.5 T R A IN P A T H S A N D T R A N S V E R S E R E C U R R E N C E

We collect together in this section several results needed later on trainpaths in a transversely recurrent train track. Let us fix a surface F = F*, where, as usual, 2g —2-f-s > 0, and if g = 0, then s > 3. By the Uniformization Theorem (see for instance [A]), there is a complete Riemannian metric of constant Gauss curvature -1 on F , which is called a hyperbolic structure on F; thus, the universal cover of F is metrically identified with the up­ per half-space IH2 = {z = x -f y \ f ^T : y > 0 } with its hyperbolic metric ds = ^ | dz |. The group of orientation-preserving isometries of IH2, called the Mobius group, is naturally isomorphic to the group PSL(2,IR) of twoby-two matrices with real entries and determinant one, where we identify each such matrix j with —7 ; thus, the fundamental group 7Ti(F) of F is identified with a (conjugacy class of) discrete subgroup(s) (of the first kind) in the Mobius group. Let = IR U {0 0 } denote the circle at infinity, where IR forms the frontier of IH2, and 00 is an ideal point which compactifies IR to a circle. Furthermore, let D C x denote the diagonal, and define the Mobius band Moo beyond infinity to be the quotient of x S^, —D by the fixedpoint-free involution S ^ x S ^ -P -S ^ x S ^ -P (*,y) ^ (y>*)Unoriented geodesics in IH2 are in one-to-one correspondence with points of Moo, where the class of a pair (z, y) E x corresponds to the geodesic g in IH2 which is asymptotic to z and y. The points x ,y are called the ideal points of g, and we define E(g) = {z,y} E Moo- Finally, the action of the group of Mobius transformations on induces an action of this group on Moo, and in particular, there are induced (continuous) actions of 7Ti(F) on and Moo • Finally, throughout this section, r C F will denote a transversely recurrent train track, F is given some fixed hyperbolic structure, and f C IH2 denotes the full pre-image of r. 60

T R A IN PATHS A N D T R A N S V E R S E R E C U R R E N C E

61

P ro p o s itio n 1.5.1: Suppose that r is a transversely recurrent train track and p is a semi-infinite trainpath on f . There is a unique point x(p) E so that p meets every neighborhood of x(p) in IH2 USj0. I f p' C p is a semi­ infinite sub-trainpath on f , then x(p) = x(p'). Finally, if p is a bi-infinite trainpath on t and p i,p 2 C p are disjoint semi-infinite sub-trainpaths, then x (pi) ¥ x (pi)P roof: Since r has only finitely many branches (say, by Corollary 1.1.3), there is some branch e of r so that the projection p of p to a trainpath on r traverses e infinitely many times. By transverse recurrence of r, there is some connected multiple curve C which meets e. We may apply an isotopy of the identity map F —►F to r as well as to C to arrange that C is a geodesic and consider a lift C of C to IH2. C cuts IH2 into two hyperbolic half-planes, and the pair E (C ) of ideal points of C cut the circle SJq at infinity into two open intervals. We claim that once p enters a component of IH2 —C , then it cannot escape: if p meets C in more than one point, then there is an imbedded bigon in IH2 whose closure in IH2 is compact and whose frontier is contained in pUC. It follows that this bigon has its frontier in fU C, and this contradicts that C hits r efficiently by Proposition 1.3.2. Since p traverses e infinitely many times, it follows that p meets an infinite number {Ci : i > 1 } of lifts of C, where Ci+\ denotes the next lift of C encountered after (7, in the natural (prescribed) orientation of p. Let I i C denote the closure of the interval with endpoints E ( C i ) which contains E ( C i + \ ) , and suppose (by conjugating by an isometry of IH2, if necessary) that I\ C IR C is a finite interval with h D h D __ By compactness of I \ , the intersection of all these intervals is non-empty. By discreteness of the action of ^i(F ) on IH2, the Euclidean diameters of these intervals tends to zero, so their intersection is in fact a singleton ;>i and insofar as p meets each neighborhood (in IH2 US^0) of x by construction, this point x(p) = x is actually independent of our original choice of multiple curve C . The assertion in the proposition about a semi-infinite sub-trainpath pf C p holds by construction, and it remains only to prove that the points x(pi) and x(p 2) determined respectively by two disjoint semi-infinite subtrainpaths pi and p 2 of a bi-infinite trainpath p are distinct. To this end, if C is any lift to IH2 of C which meets p, then E ( C ) decomposes S;^ into disjoint open intervals one of which contains x{p\) and the other of which contains x(p 2), so x(pi) ^ #(/>2)q.e.d. R e m a rk : From Theorem 1.4.3, it follows immediately that in the appro­

62

S E C T I O N 1.5

priate hyperbolic structure, each trainpath on r is a A'-quasi-geodesic, for some K depending only on L q and eo- In other words, each trainpath is A'-Hausdorff close to some bi-infinite geodesic in the hyperbolic plane. Proposition 1.5.1 follows immediately. If p is a semi-infinite trainpath on f, the point x(p) is called the limit point of p. It follows from the previous proposition that a bi-infinite train­ path p on f determines a pair x, y of distinct limit points in . Let p stand for p U{x, ?/}, and let the class of {x, y} in the Mobius band Moo be denoted E{p). Conversely, such a point of Moo essentially uniquely determines the trainpath as we show in Corollary 1.5.3 below. P ro p o s itio n 1.5.2: Suppose that pi and p 2 are each trainpaths on f, where r is transversely recurrent. Then there can be no imbedded bigon in M 2 U SJq whose frontier is contained in pi U p 2 P ro o f: We first remark that there can be no imbedded bigon in IH2 with frontier in p\ U p 2 whose closure is compact by Corollary 1.1.2. Next, we claim that there can be no imbedded bigon B in IH2 U with frontier in pi U p 2 so that one vertex, say p, of B lies in IH2 and the other vertex, say q, lies in . For if so, choose an innermost such bigon, conjugate by an isometry of IH2 (if necessary) so that q = oo, and let c l , c r be the branches of f which are incident on p and lie in the frontier of B. Index these branches so that when traveling away from p in B, e^ lies to the left and eR to the right, and let and c*r, respectively, denote the semi­ infinite trainpaths corresponding to the frontier edges of B containing e i and eR. Define two new trainpaths in f starting from p as follows: /3l begins with branch and ever after takes the rightmost possible branch at each switch (like the “righthand rule” for escaping from a maze); similarly, (3r begins with eR and ever after takes the leftmost branch. See Figure 1.5.1a. By Corollary 1.1.2 again, the trainpaths —e^, — — eR are all pairwise disjoint (except for the obvious initial points), so /?£ and f3r each have oo as limit point. Thus, /3l U {oo} and (3r U {oo} together bound an imbedded bigon B ' in IH2 U with vertices p and oo, and by construction, there is no trainpath in f fl B' from p into where B ' denotes the closure of B ' in IH2. It follows that any trainpath in f H 5 ' which meets B' gives rise to either an imbedded monogon with compact closure in IH2, in contradiction to Corollary 1.1.2, or to an imbedded monogon with oo as vertex; see Figure 1.5.1b. Thus, either f fl B' — 0, or there is an imbedded monogon in IH2 U Sjo with frontier in f and ideal vertex, and these possibilities are not tenable, as we next see.

TRA IN PATHS A ND T R A N S V E R S E R E C U R R E N C E

6R

aL

63

R

aR

(b )

(a) F IG U R E 1.5.1

In the first case, any covering translation 7 £ tti (F) must either map B' off itself or preserve it setwise (as in the last step of the proof of Proposi­ tion 1.3.2). However, y(B') = B l implies that 7 (p) = p, so 7 is the identity map. It follows that the covering projection is an embedding on 5 , which is seen to contradict local finiteness of r. In the second case, the monogon could project only to a smooth annulus with frontier in r or to an imbedded monogon component of F —r, and neither case is possible. Finally, we prove that there can be no imbedded bigon in IH2 U with frontier in pi U p 2 both of whose vertices lie in S ^ . Using the righthand rule as before leads us to either an earlier case or to the possibility that this bigon is imbedded in IH2 U Sjo —f. Such a bigon could project only to a smooth annulus or to an imbedded bigon component of F — r, and neither case is possible. q.e.d. C o ro llary 1.5.3: Suppose that p\ and p2 are each bi-infinite trainpaths on t, where r is transversely recurrent. I f E(p\) = E(p 2), then p\ is equivalent as a trainpath to either p2 or the reverse of p2 . P ro o f: It follows from the previous result that if i£(pi) = E(p 2), then pi and p 2 have exactly the same image in IH2. One easily produces the required equivalence of trainpaths, and the proof is complete. q.e.d.

64

S E C T I O N 1.5

For any transversely recurrent train track r, let us define E(r) = {E(p) : p is a bi —infinite trainpath on ?} C MooOur main result for this section is T heorem 1.5.4: I f r C F is a transversely recurrent train track, then E{r) is closed in the Mobius band beyond infinity. Proof: Let &i,... ,6n denote the branches of r. Since r is transversely recurrent, we may apply Corollary 1.3.5 to conclude that there is some multiple curve C C F hitting r efficiently so that C fl &,• ^ 0 for each i — 1, . . . , n , and, as in the proof of Proposition 1.5.1, we may assume that each component of C is a geodesic. Let us select for each i a point Pi G b{ fl C , and let C* denote the component of C with pi £ C{. Choose a point po £ r and a lift po of po to IH2 once and for all. Consider a trainpath a:[—m ,m \ —* f, m > 1 , with po £ a( ( — If the first and last branches of a are lifts to IH2 of branches 6; and bj of r, respectively, then a determines unique lifts Ci of Ci and Cj of Cj in the natural way; namely, Ci and Cj contain the lifts of pi and pj to the first and last branches of a, respectively. Each of Ci and Cj divides IH2 into two hyperbolic half­ planes, say H f and , respectively, where the superscripts are chosen so that the (oriented) trainpath a passes from H~ to H f and from H J to H'j . Let lf~ and 1^, respectively, denote the closed intervals in which form the ideal boundary of H f and H^r, and define Jm(po) = U

K 'r X I j ) U ( I f X / f )] C S ^ X Sjo,

where the union is over the set of all trainpaths a: [—m,m] —►f so that G a (( —~ ,|) ) , as above; this is actually a finite union of distinct closed sets, hence J m(po) is itself a closed subset of x S ^ , for each m > 1 . Thus, the intersection

Po

J(Po)

=

P i Jm(Po)

m> 1 is a closed subset of x S ^ . Furthermore, this intersection is nested J m+ i(p0) C Jm{Po)y and the limit points of any bi-infinite trainpath on f which passes through po are contained in J m(po), for all m > 1 . It follows easily that J(p0) = {E(p) : p is a bi —infinite trainpath on f with po G p]

T R A IN PATHS A N D T R A N S V E R S E R E C U R R E N C E

65

is a closed subset of x S ^ , for any p 0 E f. In fact, J(po) C x S ^ —D by Corollary 1.5.3 (and J(po) is saturated for the projection x —D —► Mqo by construction), so J(po) descends to a closed subset of Moo itself. Now, suppose that E(pi) = {xi,Pi} E E ( t ), for i > 1, where {/?,} is a collection of trainpaths on f and {xt-,y,-} converges to {x,y] E Sjo x S Iq—D in the topology of the Mobius band beyond infinity. We may assume that the pi are oriented with initial limit point Xi and terminal limit point 2/,in such a way that Xi converges to x and yi converges to y. If there is a point of f through which pass infinitely many pi, then the argument given above applies to show that there is a bi-infinite trainpath on f with ideal points #,y, and the theorem follows in this case. We may thus assume that through any point of f pass only finitely many pi\ let us call this “supposition (*)” . We assume for now that y £ {?/,}, x £ {#*•}, and claim that the ideal points Xi,pi cannot separate x from y in for any index i. Suppose, to the contrary, that x j , yj do separate x from y , so that infinitely many trainpaths pi must intersect pj. On the other hand, by supposition (*), for any finite sub-trainpath /? C Pj, there can be only finitely many trainpaths among {pi} which contain points of /?. As in the proof of Proposition 1.5.1 (by discreteness of the action of ni(F) on IH2), since x ^ y, there are lifts Cx and Cy to IH2 of components of the multiple curve C above (which hits r efficiently), so that each of Cx and Cy meets pj (necessarily in a singleton by Proposition 1.3.2), the ideal points E(CX) separate xj from x , and the ideal points E(Cy) separate yj from y. Let ft C pj denote the smallest finite sub-trainpath of pj containing pj C\{Cx \JCy). If pi meets pj —/?, then E(CX) (and E(Cy), respectively) must separate Xi from x (and yi from y). This contradicts the convergence of Xi to x and yi to y, and we have established that no pair Xj, yj can separate x from y in . Next, we observe that no two trainpaths on f can bound an imbedded bigon in IH2 with compact closure by Proposition 1.5.1 (or Corollary 1.1.2), so, by passing to a subsequence, we may arrange that for any trainpath pj in our sequence, every other trainpath in the sequence lies in the closure of the component of IH2 U Sjo —pj which contains {x,y}. The pair {x , y } decomposes into two closed intervals 7, V , and if we can find E(pj) C 7 and E(pk) C 7', then argue as follows: choose an arc S imbedded in IH2 connecting pj and pk, and consider an infinite sequence defined by Z{ E pi H 6 if pi D 0, there is a 8 > 0 such that if q C IH2 U is a K -quasi-geodesic whose ideal points lie within 6 of the pair x ,y (in the Euclidean metric on IH2 U S^,), then q must stay within K + c of g (in the hyperbolic metric on IH2). This is a contradiction, because for all sufficiently large i, the ideal points yi must be within 6 of x, y yet must miss an appropriately chosen D (for instance, a disk D of radius greater than I\ -f e). As a final note, we add P ro p o s itio n 1.5.5: I f t \ , t 2 C F are transversely recurrent train tracks so that T\ < 72 (i.e., r2 carries t\), then E{t\) C E ( t 2). P ro of: Let f\ and T2 , respectively, denote the full pre-images of T\ and r 2 in IH2. If — 0. In­ deed, keeping the normalizations and notations of the previous paragraph, consider a component H ' of the full pre-image of Up», so H' is the interior of some Euclidean disk tangent to IR C S ^ . If H fl H' ^ 0, then this disk must have diameter exceeding unity, so H 'n y ( H ') ^ 0 , and this contradicts that Upi is a deleted neighborhood of pf in F. It follows that WUA is indeed a regular neighborhood of A in F, and it remains to show only that if a geodesic lamination G meets 7/, then G does not have compact support. Suppose that a leaf g of G meets a component, say 7/p, ofU. Again we adopt the normalizations and notations of the earlier paragraph, and consider a lift g of g to IH2 so that g fl H ^ 0. If g does not meet every neighborhood of p, then g is a Euclidean semi-circle which is perpendicular to IR C S ^ , and since g meets 77, this semi-circle must have radius exceeding unity. It follows that gC\j(g) ^ 0, contradicting that g is simple. q.e.d. R e m a rk : The proof above that Up is imbedded is borrowed from [A] and

73

LAMINATIONS

included for completeness. We say that a lamination (or geodesic lamination) L in F is carried by a train track r C F if there is a supporting map :F —►F satisfying the same three conditions as in the definition of carrying in §1.2 (where L is substituted for the multiple curve C) and write r > L } as before, if r carries L. T h e o re m 1.6.5: Every geodesic lamination G in F with compact support is carried by some transversely recurrent train track r C F. P roof: Arguing as in the proof of Proposition 1.6.3 above, we may add a finite collection of geodesics in F to G to produce a geodesic lamination G \ whose full pre-image in IH2 is denoted G', so that each component of IH2 — G' is either an open ideal triangle or perhaps the interior of an infinite-sided ideal polygon whose vertices comprise the orbit of some point of SJq under a cyclic parabolic subgroup of 7ri(F); each component of the latter type projects to a once-punctured region in F whose frontier consists of a single geodesic in G', and as before, each component of the former type imbeds in F . Such an extension is possible since G has compact support, s o G c f - W b y Proposition 1.6.4, and the resulting geodesic lamination G* has compact support as well The closed e-neighborhood Ne of G' in IH2 has piecewise C 1 boundary dNe with one cusp (i.e., non-smooth point) for each pair (R, v), where R is a component of IH2 —G' and v is an ideal vertex of R. Ne projects to the e-neighborhood N e of Gf in F , and the boundary dNe is piecewise C 1 with only a finite number of cusps; indeed, the number of cusps is given by a constant K which depends only on the topological type of F; namely, K = —6x(F ) —2s as in the proof of Proposition 1.6.3 (where, as usual, s denotes the number of punctures of F).

(a)

(b) F IG U R E 1 .6 .2

Since F —G 9 has a finite number of finite-sided components by Propo­ sition 1.6.3, G* has only finitely many (disjoint) frontier leaves, and we may

74

S E C T I O N 1.6

choose e > 0 with the following property: if g and g' are frontier leaves of G' arising from the same component of IH2 —G', then the hyperbolic distance from g to gf is less than 3£ if and only if E{g) fl E(gf) ^ 0. Thus, for any component R of IH2 —N e, there is a component R of IH2 —G' so that each frontier edge of R lies in an £-hypercycle of the corresponding frontier geodesic of R ; see Figure 1.6.2a and b. (Recall that an “e-hypercycle” h£ about a hyperbolic geodesic g is a component of the frontier of the metric £-neighborhood of g. The two key fact for us is: the component of IH2 —h£ containing g is convex. ) Next, we claim that there is some 6 > 0 so that given any two points Pi Q € G' that are within 6 of one another, the tangent lines to the leaves of G' through p and q are close. To see this, suppose that g\,g 2 are distinct geodesics in G' and pi E 0 if and only if a H L / 0. If A is a measure on the lamination L, then the pair (L, A) is called a measured lamination, and if L is actually a geodesic lamination, then the pair is called a measured geodesic lamination. In practice, we will sometimes suppress A from the notation and refer to L itself as a measured (geodesic) lamination. The simplest example of a measured geodesic lamination (G , A) is when G is a collection {C j}j^ j of geodesic curves disjointly imbedded in F, each curve Cj is given some “weight” jij £ IR+, and the transverse measure A of a £ A(G) is given by A(a) = ^ / i j card (a D C j ).

82

M EASURED LAMINATIONS

83

If (G,A) is a measured geodesic lamination on F and G is the full pre-image of G in IH2, then G determines a closed, 7Ti(F)-invariant subset E (G ) C Mqo as in Corollary 1.6.2. The transverse measure A furthermore determines a 7Ti (^-in v arian t measure Z \ on Mqo with support E (G ) as follows. Let p be a point of G and choose a neighborhood B of p in IH2. There is an identification L, and a preliminary endeavor is the construction of a certain useful bi-foliated neighborhood of r in F; in the second step, we “straighten” L to an associated geodesic lamination G < r, and it is in this step that transverse recurrence of r is indispensible; the third and final step is the assignment of a transverse measure to G.

R e m a rk : W ithout transverse recurrence of r, there is no guarantee of a metric on F in which L may be straightened to the geodesic lamination G. We will see in Theorem 2.7.4, however, that arbitrary measured train tracks give rise to measured geodesic laminations because any measured track may be “refined” to one which is transversely recurrent, and we may apply Construction 1.7.7 to the refined track.

We may assume that /i > 0, for, if not, then

t'

— Tp — t — {branches b of r: fi(b) = 0 }

is a subtrack of r (where we amalgamate each appropriate resulting bivalent switch), and the original measure /i on r determines a positive measure \i* > 0 on r'. Since r ' C r, r f is transversely recurrent by Lemma 1.3.3a. Applying Construction 1.7.7 to the measured track (r',//') then associates a measured geodesic lamination to (r, fi). S te p 1: Enumerate the branches of r once and for all, and consider a collection of abstract (oriented) Euclidean rectangles with the same index set; the rectangle Ri is taken to have length one and width /x(6,-), and we choose, once and for all, a vertical segment £; in the middle of i?*, for each i. Each R{ is endowed with two canonical foliations Ti (and F /1, respectively) with leaves perpendicular (parallel) to and the boundary dRi decomposes into two “horizontal” arcs (which are perpendicular to £*• and of length one) and two “vertical” arcs (which are parallel to & and of length fi(bi)); see Figure 1.7.2. Each branch 6; of r has exactly two ends, and we may index the vertical arcs in dRi by the ends of &*; each such vertical arc inherits an orientation from that of the rectangle Ri.

MEASU RED LAMINATIONS

89

FIG U R E 1.7.2 We define a bi-foliated surface which is naturally homeomorphic to a closed regular neighborhood TV of r in F by identifying certain sets of vertical arcs in the boundaries of rectangles as follows. Suppose that v E F is a switch of r with e i , . . . , er the incoming ends of branches incident on v and er + i , ... , er +* outgoing. We may suppose that the indices are chosen so that as one traverses a small simple closed curve in F encircling v (in the clockwise direction), one encounters the half-branches of r corresponding to the ends 6^,..., ,..., m this ordor, let ct^,..., flj>, j,..., denote the corresponding oriented vertical arcs. By the switch conditions, the sum C of the Euclidean lengths of agrees with the sum of the Euclidean lengths of {ajYjtl+n and there are unique Euclidean-lengthpreserving maps r

f - L la* i=i and r+t

g: J J dj j=r+ 1

— *

[0,C],

where / is orientation-reversing and maps the initial point of a\ to £, g is orientation-preserving and maps the endpoint of ar + 1 to C, f(dai) D f ( d a i+x) ^ 0, for i = 1, . . . , r - 1, and g(daj) H g(daj+i) ^ 0, for j = r + 1, . . . , r + t - 1. Finally, each fiber of the map / J j # is collapsed to a (distinct) point; see Figure 1.7.3.

90

S E C T I O N 1.7

FIGURE 1.7.3 Take the quotient of R{ by these identifications as v ranges over all the switches of r. The result is a surface which we henceforth identify with a neighborhood N of r in F in order to simplify notation. The fo­ liations and {^YL}f_1 combine in the natural way to produce the transverse foliations T and T 1' , respectively, and each of T and T L inherits a transverse measure (in the sense of [FLP]) from the Euclidean metrics on the rectangles. A point p G d N is called a singular point if the fiber of the natural projection U ”=1 Ri —►N over p has cardinality greater than two; the sin­ gular points are thus the “cusps” of dN. A leaf of T L is a tie, and it is a singular tie if it moreover contains a singular point. If E is the union of all the leaves of T which contain singular points and II is the set of all singular points, then each “component” of E —II is called a singular leaf of T , and any leaf of T other than a singular leaf is a non-singular leaf. If I is a leaf of T which is not a simple closed curve and p G t, then each component of £ —p is called a half-leaf of. t. Thus, any half-leaf of a non-singular leaf has infinite T L-measure. In contrast, a singular leaf is called finite if every half-leaf of it has finite -measure and is called semi-infinite otherwise. Notice that if I is a semi-infinite singular leaf and p € £, then exactly one component of £ — p has finite F 1 -measure. Finally, notice that each sin­ gular leaf must lie either entirely in the interior of N or entirely in ON] furthermore, a semi-infinite singular leaf is always of the former type while a finite singular leaf might be of either type. This completes the required construction of our bi-foliated neighbor­ hood (A, F, F 1 ) of r in F. (This neighborhood is technically useful both in the rest of this section and elsewhere in this work.) The idea of the construction of a lamination L associated to (JV, F, F L ) is now easy to de­ scribe: alter the foliation F by “splitting” along all of the singular leaves, so that only “packets” of parallel non-singular leaves remain. These leaves form the lamination L. To make this precise, let 0a- denote the intersection of & with the col­

91

M E ASURED LAMINATIONS

lection of all the singular leaves of T , and notice that 9{ is either a finite or a countably infinite set (and 9% — 0 if and only if bi corresponds to a curve component of r). For each i = 1, . . . , n, we will construct an inter­ val rji C IR together with a closed subset Ki C rji and a continuous map H'-li & so that the following conditions hold. (i) Letting m denote the Lebesgue measure on IR, we have m(Ki) = ji(bi) = m(&), and m(rji) < 2m(&). (ii) The restriction Li \x t is one-to-one over points of

U(& —9i).

(Hi) The restriction ti \k { is two-to-one over points of 0,- — To construct such intervals and maps, i x ij}]Li (where perhaps rii = oo), and let given by adding to the Lebesgue measure m 2" Jm((,') at Xij, for each j — 1, . . . , n,*. For interval of length

enumerate the set 9i C & as m; denote the measure on each Dirac measure with mass each i — 1, . . . , n let rji be the

nt

m(r]i) = m (& ){l + ^ 2 ~ J } < 2m(&). j= 1 The distribution of maps & to rji and is discontinuous exactly on 9i. The inverse distribution tf. rji —>&, however, is continuous. Let Ki C rji be the complement of the interior of the intervals (which map to 0,*) on which ti is constant. It is evident that the three demands above are met. Fix a switch v, and recall the intervals and maps r r+t f- L I ai -*• [0,C] and g: J J a, -* [0,C], i= 1 j=r+ 1 which arose in the construction of the neighborhood N. Now consider a new collection of abstract Euclidean rectangles {#(•}”_! where R\ has length one and width 2/i(6;). Let a[ y. . . , a'r , aj.+ 1, . . . , a'r+t denote the corresponding vertical arcs in and r

r+t

f : | J a'- -*■ [0,2respectively. A leaf of which contains a point of Q(m) —Q(ra) is called a “singular tie” of T ^ . As in the proof of Theorem 1.6.5, collapsing each leaf of to a point produces a train track rm C Nm , where the singular ties of give rise to switches of rm: see Figure 1.74b. Using this collapse, one easily shows that rm > rm+1. At the same time, the collapse of ties of itself produces the original track r, and it follows similarly that t = tq > rm) for each m > 1 .

M E ASURED LAMINATIONS

95

Since 7o is transversely recurrent by hypothesis and, moreover, To > rm, Lemma 1.3.3 part (b) applies to show that rm is also transversely recurrent, for each m. It follows that each E(rm) C Moo is closed by Theorem 1.5.4, and since rm > rm+1 , we find that E(rm) D E(rm+1), for m > 0, by Proposition 1.5.5. The intersection £ = f l E(rm) C Moo m> 0

is therefore nested, and £ is a closed subset of Moo- Each rm carries L by construction, so E(rm) D E(G) by definition, and it follows that E(G) C £. To prove the reverse inclusion requires some work. It is convenient to introduce the following notion: suppose that a: [0,7] —* N for 7 £ ZZ+, is an arc in N which is transverse to the ties of T L so that each point a(i), for i £ [0,/] fl 2Z, lies in a singular tie of , and these are the only intersections of a with singular ties of . Such a path a is called a (finite) tie-transverse path in N of length 7, and two tie-transverse paths are regarded as equivalent if they are homotopic along in the sense that there is a homotopy from one to the other so that the track of any point (under the isotopy) is contained in some tie of N. There is a natural one-to-one correspondence between the set of all (equivalence classes of) tie-transverse paths of length 7 in TV and the set of all (equivalence classes of) finite trainpaths on r of length 7; this correspondence is induced by the correspondence between the branches of r and the component rectangles of TV.

The following result gives the crucial estimate in the proof below that £ C E(G) and is of interest in its own right. L em m a 1.7.9: Suppose that p is a trainpath on r of length at most 2m + 1 and the tie-transverse path of p is homotopic along the leaves of to a path contained in Nm, for some m > 0. Then p is realized by a leaf £ of L in the sense that p arises by restricting the domain of a trainpath on r corresponding to £. P roof: By definition of the lamination L, it suffices to prove that p is realized by some non-singular leaf £ of T . Let the length of p be j, where 1 < j < 2m + 1, and proceed by induction on j. For the basis step j = 1 , suppose that 6 is a branch of r and let f be a tie of T 1 meeting 6. Since the intersection 9 of £ with the collection of all singular leaves of T is countable, we must have £ —6 ^ 0 . Simply choose any leaf £ of T meeting £ —9 to complete the basis step. For the inductive step, pass to the universal cover IH2 of F and con­ sider the full pre-image in IH2 of the bi-foliated neighborhood

96

S E C T I O N 1.7

(N, T , Jr±). Let p: [0, j] —» N be a lift to IH2 of a trainpath p on r of length j , where 2 ra-f 1 > j > 2 , so that p |[o,j-i] and p l y-i j], respectively, are re­ alized by leaves £\ and £2 of T by the inductive hypothesis. Let £\ C t\ and 4 C respectively, denote lifts of the corresponding finite tie-transverse paths so that the terminal point p of £[ and the initial point q of £'2 lie on the same singular tie to of . Suppose, for definiteness, that £[ lies to the left of p and £^ lies to the right of q near to. There may be singular points of T on to between p and q, and we distinguish two cases: such a singular point s 0 is called a “right cusp” if the singular leaf of T issuing from s 0 lies to the right of t 0 near so; So is called a “left cusp” in the contrary case. (See Figure 1.7.5a.)

to

to

rights cusp

-n* - --------£l P

— —

■ -

ip

Qh jLi

-left 'b1. cusp £2

t 2(b )

(a)

^b

'b

'b ff'0.

q

to

tl

to a. Of p "STT

'b -C4-

O,

£2

(c)

Or ^IT (d)

FIGURE 1.7.5 We finish the proof of the lemma by induction on the number c of left cusps on to between p and q. If c = 0, then £\ actually realizes p; see Figure 1.7.5b. It follows that a collection of nearby leaves also realize p, and the proof is completed by a cardinality argument as before. If c > 1, let so G ^0 denote the left cusp nearest p lying between p and q on to, and let 0. It follows that for each m, the restriction po |[_m m+1]

98

S E C T I O N 1.7

satisfies the hypotheses of Lemma 1.7.9, and the corresponding leaf gives rise to a point of Jm . Thus, Jm ^ 0 for all m > 1, and this completes the proof that £ C E(G). It follows that £ = E(G) C is closed, so G is a geodesic lam­ ination. Finally, since E(G) = £ C E(rm), Theorem 1.6.6 implies that G < rm, for all m > 0. In particular, G < To = r, and G has compact sup­ port by construction, so the proof of Theorem 1.7.8 is completed. q.e.d. We record here the following immediate corollary of the argument above. C o ro llary 1.7.10: Suppose that (r, p) is a birecurrent train track giving rise to the geodesic lamination G C F and a C N is a bi-infinite tie-transverse path which is disjoint from the singular leaves of J7. I fp denotes a trainpath on the full pre-image t C IH2 corresponding to a, then E(p) E E(G). This finishes our discussion of the geodesic lamination G, and Step 2 is complete. S te p 3: It remains to define a transverse measure A' on G, and this is easily done. Let A' denote the collection of arcs transverse to G with endpoints in F — G. We claim that any arc a* E A' can be written as the concatenation ol

— a[ * #2 * ... * a K ' ,

where each arc a k meets the lamination G efficiently in the sense that there is no bigon imbedded in F whose frontier consists of one C 1 segment contained in a fk and one contained in a leaf of G. To prove the claim, we suppose the contrary, so there are infinitely many disjoint sub-arcs of a which do not meet G efficiently; these sub-arcs have an accumulation point, say x E a ', and x is necessarily a point of a* flG since each sub-arc contains a point of a! fl G, and G is closed. This violates transversality of a' with G at the point x and proves our claim. Thus, by Property (2) in the definition of transverse measure, it suffices to define A' on the arcs in A' which meet G efficiently. Recall the lamination L constructed in Step 1 together with its transverse measure A defined on the family A of arcs. An efficient arc a f E A' gives rise in the natural way to an arc a E A which meets L efficiently, and we define A'(a') = A(a).

The required properties of A' follow directly from the corresponding prop­ erties of A.

MEASU RED LAMINATIONS

99

This finishes the construction of a transverse measure A' on G, so Step (3) is complete. Thus, we have described the construction of a mea­ sured geodesic lamination (G, A') from a measured recurrent train track To finish our discussion, we prove that Construction 1.7.7 is actu­ ally the two-sided inverse of the construction of the measure A*(G,A) on T from (G, A) when G < r, thus canonically identifying the collection of all measured geodesic laminations of compact support carried by r with the collection of all measures on r. L em m a 1.7.11: Suppose that r C F is a transversely recurrent train track. (a) I f fii and p 2 are measures on r giving rise to the same measured geodesic lamination (G, A), then p\ = //2. (b) //(G j, Aj) are measured geodesic laminations on F , with G( < r giving rise to the same measure pi = pi(Gi}\ t)> f or i = 1>2, then (Gi, Ai) = (G2, A2). P roof: For part (a), choose a supporting map $ for the carrying G < r, fix a branch 6 of r, and choose a point p £ b which is a regular value of . Choose an arc crossing b transversely at p and meeting r nowhere else, and let a'b = By construction and Proposition 1.7.5, p\(b) = A(a'&) = //2(6), as desired. For part (6), recall the construction of the sequence r = r0 > r x > ... of birecurrent train tracks built from the measure pi on r as in the proof of Theorem 1.7.8. As shown there E(Gi) = n m>oE(Tm) = E(G 2), where Gi denotes the full pre-image of G* in IH2. We conclude that G\ = G2, as was claimed. q.e.d. The measured lamination space M.C(F) of a surface F is the collec­ tion of all measured geodesic laminations in F , where F has some fixed hyperbolic structure. The topology on M C (F ) is induced from the weak topology on measures supported on as in the remark following Propo­ sition 1.7.1. This topology (and indeed M C (F ) itself) seems to depend on the hyperbolic structure on the surface F. However, Nielsen’s Exten­ sion Theorem says that the lift to IH2 of a continuous mapping Fi —►F2 between two hyperbolic surfaces extends continuously to S ^ ; thus, a homeomorphism of F induces a homeomorphism Mqo —► Mqo, which may be used to naturally identify M C (F i) with A4£(F2). Hence it is legitimate to

100

S E C T I O N 1.7

write simply M C (F ) for the collection of all measured geodesic laminations in F without any reference to the underlying hyperbolic structure on F . Another description of the topology of M C {F ) (which is often useful) is as follows. Suppose that (G, A) E M C ( F ), let e E IR+, and let {&k}¥ be a finite collection of arcs lying in A(G). A basic open set in M C (F ) about (G, A) is given by the collection of all (G',A') E M C (F ) so that a k E A(G') and |A(a*) - \ ' ( a k)\ < e for all k = Using the assignment (G,A) i—►E \ defined before Proposition 1.7.1, one sees without difficulty that that the two topologies defined on M C ( F ) actually coincide. We define M Co(F) C M £ ( F ) to be the subspace of measured geodesic laminations in F with compact support. Elements of M Co(F) lie outside the unit horoball neighborhood of the punctures of F by Proposition 1.6.4. Multiplying the transverse measure on a geodesic lamination by a posi­ tive constant gives an IR+-action on each of M C (F ) and A4Co(F), and this action is evidently properly discontinuous. The space of projective lamina­ tions in F is defined to be VC(F) = (M C (F ) — {0})/IR+, where 0 denotes the empty lamination, and the space of projective laminations with com­ pact support in F is similarly defined by VCo(F) = (M £ o (F ) — {0})/lR+. The topologies on V £ (F ) and VCo(F) are the quotient topologies inherited from those on A4C(F) and M C q(F) respectively. If r C F is a birecurrent train track, then the collection V ( t ) of all transverse measures has a natural topology, where two measures are close if the weights they assign to each branch are close. (The structure of V ( t ) will be further investigated in §2.1 and §2.2.) Thus, we may regard Construc­ tion 1.7.7 as a map from the topological space V(r) to the topological space M Co(F). Collecting our results about Construction 1.7.7 into a theorem for later use, we have T h e o re m 1.7.12: Fix a birecurrent train track r C F and let V(r) be the space of transverse measures on r. Construction 1.7.7 establishes a contin­ uous injection V(r) —►M Co(F) whose image consists of all the measured geodesic laminations of compact support carried by t . R e m a rk : We will find (in Theorem 2.7.4) that the hypothesis of transverse recurrence on r can be dropped. P roof: By Lemma 1.7.11, only continuity of the map induced by Construc­ tion 1.7.7 requires comment, and we use the description of the topology on M C q(F) in terms of arcs in the surface given above. Suppose that (7 ,^ ) gives rise to the the measured geodesic lamination (G, A') and let (L, A)

M E A SU R E D LAMINATIONS

101

be the corresponding measured lamination defined in Step 1 of Construc­ tion 1.7.7. Given an arc a' E A(G), we may assume that a f meets G efficiently as in Step 3 of Construction 1.7.7, so a ' gives rise to an arc a E A(L). Finally, A'(a') is defined to agree with A(a), and this latter quantity is obviously a continuous function of fi. Thus, the map induced by Construction 1.7.7 is continuous, as was asserted. q.e.d. C o ro llary 1.7.13: Suppose that G C F is a measured geodesic lamination carried by the birecurrent train track r C F. There is a sequence r — tq > T\ > ... of generic birecurrent train tracks so that E{G) — r)m>oE(Tm).

P roof: It follows directly from Theorem 1.7.12 (using the sequence of train tracks described in the proof of Theorem 1.7.8) that there is a sequence r = Tg > t [ > ..., as in the statement of Corollary 1.7.13, but the train tracks T ’m in fact fail to be generic by construction. To correct this, we take a generic track rm arising from by a sequence of combs. Evidently, T — Tq >

t[

>

T\

>T*2 > . . . ,

and the assertions about the sequence r > t i > ... follow directly from the corresponding facts for the sequence r > r[ > __ q.e.d.

§1.8 B O U N D E D SU R FA C E S A N D T R A C K S W IT H S T O P S

This section is devoted to sketching the extension of the foregoing theory for surfaces with boundary and then developing a relative version of train tracks (from [PI]) in this setting. Let Fg,r denote a smooth, oriented surface of genus g with s > 0 distinguished points, whose union we denote A, and with r > 1 smooth boundary components, whose union we denote dFg,r. As usual, we will often regard the points of A as cusps and define Fg,r = Fg'r —A; when the topological type of the surface Fg,r (or F^ r , respectively) is fixed or not important, we may call it simply F (or F). Define a train track r in such a surface F exactly as before (and adopt the attendant terminology of branches, switches, etc.), where r is required to be disjoint from dF. The basic fact is that all of the results of §1.1 —§1.7 hold in this setting as well. The two small technical distinctions (which contribute to the rationale for presenting the bounded case separately here) are as follows: one must distinguish between two types of complementary n-gon-minus-a-disk depending on whether the smooth frontier edge lies in dF; the universal cover of F is not canonically identified with IH2, but rather with a subset of IH2. The former difference is easily handled by simply formally treating each component of dF as a puncture. The latter difference is addressed by doubling F*,r along all the curves in dFg,r to produce a surface F f of type I^ + r-1 » choosing a hyperbolic structure on F*, the universal cover of F f can be identified with IH2, and F C F f inherits its intrinsic Poincare metric, where each component of dF C F* is geodesic. Furthermore, a train track in F can be regarded as a train track in F* D F in the natural way. We next briefly discuss some of the particulars of the foregoing theory in this setting. First notice thcit by Condition (3) in the definition of train track, no curve component of a train track in F can be parallel to a compo­ nent of dF; thus, the arguments of §1.1 apply with only small modifications to show that F contains a train track if and only if x(F ) < 0 and F ls n°t one of the surfaces Fo’r , where s -f r = 3. A multiple curve in F = F*'r is defined exactly as before with the the additional proviso that no component is isotopic into dF; a train track r C F carries multiple curves in F , and 102

B O U N D E D SURFACES AND TRACKS WITH STOPS

103

a measure on r uniquely determines a multicurve carried by r. A pants decomposition of F is a multiple curve C C F so that each component of F —C — dF is homeomorphic to the interior of a pair of pants, there are jV = 3°

^0

•

B O U N D E D SURFACES AND TRACKS WITH STOPS

105

Let {Kk}k=i denote the various components of d F , and choose a closed arc Qfjk C Kk which contains Kk fl E, for each k = 1 ,..., r . A multiple arc C in F is a smooth compact one-manifold imbedded in F whose boundary (if any) is contained in U£=1 0 and twisting numbers A G 2Z, for i = 1 ,..., N + r, and arrive at a relative version of Dehn’s Theorem which gives a one-to-one correspondence between 7Z(Fg>r) and the subset of ZZ^+r x TLNArV satisfying Condition (a) and the obvious analogue of Condition (b) in Theorem 1.2.1. R e m a rk : In analogy to the above, one can also give a parametrization of

106

S E C T I O N 1.8

homotopy classes of appropriate one-manifolds imbedded in F = F U A with boundary in Uj=1a:* U A by a certain subset of ZZ++r+5 x 7LN+r. There is a difference in the treatment of boundary components of F and points of A since one can isotope an arc incident on a point in A so as to get rid of any twisting. A train track r with stops is said to be recurrent if for each branch 6 of r there is a multiple arc Cb carried by r whose associated measure is positive on 6. A (transverse) measure on r is defined as before, and a train track with stops is recurrent if and only if it supports a strictly positive measure. The definition of hitting efficiently is exactly the same as before, and Proposition 1.3.3 holds verbatim, r is said to be transversely recurrent if for each branch b there is a multiple arc hitting r efficiently and meeting 6, and a track with stops which is both recurrent and transversely recurrent is said to be birecurrent. As in Lemma 1.3.3, if r is a transversely recurrent track with stops, and r ' is a track (perhaps) with stops so that either r 3 r 1 or r > r ', then r ; is transversely recurrent (as a track with stops) as well. Suppose that r C F is a train track with stops and v is an assignment of nonnegative real number to each branch of r. v is called a tangential measure on r provided that Conditions (1) and (2) hold as before on each component of F — r whose frontier is disjoint from d F , and v is said to be even if it is [2Z+ U {0}]-valued and satisfies the evenness condition as before on each such component of F — r. In anology to Lemma 1.3.4, we claim that if v is an even tangential measure on r C F, then there is a multiple arc C C F hitting r efficiently so that C intersects each branch 6 of t in exactly v(b) points. To see this, we may add curve components to r to produce a train track with stops so that each complementary region is one of the following: i) an m-gon, where m > 3; ii) a once-punctured-mgon, where m > 1; in) an m-gon-minus-a-disk, where m > 1; iv) a pseudo pair of pants; v) an annulus with one frontier edge in dF. The smooth frontier curves in cases Hi) and iv) may be taken to be disjoint from d F } we may assume that each component of type ii) or Hi) satisfies m = 1, as before, and one easily produces an even tangential measure vn on the resulting train track r" with stops which extends the tangential measure v on r. Choose vn(b) distinct points on each branch b of r" and connect these points exactly as before in each component of F —r " whose frontier is disjoint from dF. Since a half-branch of r" incident on a stop meets d F transversely, a component of F —r" whose frontier meets dF must be of type i) or type v). If R is a component of the latter sort whose frontier edge in F —dF has total i/'-tangential measure £, simply choose £ points in the other frontier edge of f?, and then connect the two frontier edges of R with £ arcs properly imbedded in R as in Figure 1.8.2a. Consider an m-gon component of the former sort, and proceed by induction on m > 3. For the

B O U N D E D SURFACES AND T RACK S WITH ST OPS

107

basis step m = 3, notice that exactly one frontier edge, say 5, of R lies in 5F, let and £2 , respectively, denote the total i/'-tangential measures of the other frontier edges, say 8\ and 0, there is an intrinsic Poincare metric with spikes on F so that r has geodesic curvature less than e at each point, and each branch of r has length at least L. The proof of the equivalence of Conditions (i) and (ii) above is analogous to the proof given before. To see that Condition (ii) implies Condition (Hi) above, simply apply the arguments given before to F —U ^ e R ? and then adjoin a col­ lection of ideal triangles, one triangle for each fi(b) (so fi(d) > /i(c) by the switch conditions), and we perform a right split on e; /i(a) = pi(b) (so /i(c) — V>(d)), and we perform a collision along e; fi(a) < fi(b) (so fi(d) < //(c)), and we perform a left split along e. Formally introducing the variable X £ {ii, L} and supposing that r'x is determined by the measure pL as above, observe that there is a natural oneto-one correspondence between the set of branches of r and those of rx ; if / is a branch of r, then let f x denote the corresponding branch of rx as indicated in Figure 2.1.2b. There is a measure ptx induced on rx according

120

S E C T I O N 2.1

to the rule u> ( fl \ _ f !/* (« )-M&)l> i i f ' x = eX ’ x \ /*(/)> otherwise. Of course, two branches of t typically amalgamate to form each of the branches a!Q and d'Q of r f0 which are indicated in Figure 2.1.2b. To any branch f fQ of Tq other than a'0 or d!0 , there is a corresponding branch / of r, and /i induces a measure fi‘0 on t ’q according to the rule p'oU'o) = M/)>

if flj is a branch of r^ . Thus, in each case, the measured track (r, //) uniquely determines a measured track ( r ;, ^/') = (Tfx ,fifx ), for the appro­ priate X E {i?, O, L}, and we say that (r', fif) arises from (r, p) by a spW along the branch e. Furthermore, inverting the formulas above, if r arises from r ' by a collapse, then a (positive) measure on r ' gives rise to a (positive) measure // on r; and we say that the measured track (r, //) arises from (r', /i') by a collapse. As before, we remark that recurrence is invariant under both splitting and collapsing measured tracks. L em m a 2.1.3: Suppose that r is a recurrent track and let r'R, Tq , and r'L, respectively, be the tracks produced from a right split, a collision, and a left split along some large branch. Either all three tracks r'R, Tq , and t *l are recurrent, or exactly one of them is recurrent. P ro o f: Since r is recurrent, it supports a positive measure //, and (r, //) splits to the measured track (Tx ,pifx ), for some X E {/2,0,L }, with nx > 0; thus, at least one of the tracks rR, r^, and r'L is recurrent. To complete the proof, we must show that recurrence of two tracks among rRl r '0 , and t'l implies recurrence of the third. Since a train track is recurrent if and only if each connected component is recurrent, we may assume that r (and hence also rR and t'l ) is connected. Finally, we adopt the notation of Figure 2.1.2b for the branches of r, rR , t '0 , and r'L . We begin by showing that recurrence of rR and t '0 implies recurrence of r£, and, insofar as Tq C t 'l is recurrent, t '0 lies in the maximal recurrent subtrack of r fL ; to prove the implication, we must therefore produce a closed trainpath on r fL which traverses the branch e'L . If t 'q is connected and nonorientable, then by Basic Fact 2 of Proposition 1.3.7, there is a trainpath on Tq which begins by traversing a!Q and ends by traversing d!Q with the respective orientations indicated in Figure 2.1.3a. This trainpath gives rise to a corresponding trainpath on D Tq , and it is an easy m atter to construct the required closed trainpath on t 'l .

SPLITTING, SHIFTING, AND CARRYING

(a )

(b )

121

(c)

F IG U R E 2.1.3 If Tq is connected and orientable, then there are two cases depending on the relative orientations of a'0 and d!Q. If these orientations “agree” as in Figure 2.1.3a (or if both orientations are reversed), then by Basic Fact 1 of Proposition 1.3.7, there is a trainpath on t 0' which begins by traversing afQ and ends by traversing d!Q (in their indicated orientations), and one produces the required closed trainpath on t 'l as before. If the orientations on a!0 and d!Q “disagree” as in Figure 2.1.3b (or if both orientations are reversed), then it follows that rR is non-orientable. By Basic Fact 2 of Proposition 1.3.7, rR must support a trainpath which begins and ends by traversing eR with opposite orientations, and this is clearly impossible; thus, this second case is not tenable. To complete the proof of the implication, we must finally consider the possibility that t 'q is disconnected. In this case, recurrence of rR implies that each component of Tq is non-orientable. Orient a!0 and d!Q as indi­ cated in Figure 2.1.3b, apply Basic Fact 2 of Proposition 1.3.7 to produce a trainpath on r l0 which begins by traversing a'0 (in its prescribed orienta­ tion) and ends by traversing the reverse of a!0 , and construct an analogous trainpath on Tq traversing d^. These trainpaths give rise to corresponding trainpaths on t l => *o> and these combine with two trainpaths of length one on t'l which traverse e*L (in opposite directions) in the natural way to produce the required closed trainpath on r fL . Arguing in analogy to the above, one finds that recurrence of Tq and r fL similarly implies recurrence of rR , and it remains only to show that if rR and t'l are recurrent, then rl0 is recurrent as well. If t'0 is connected, then it is evident that at most one of t r and is orientable, and we may suppose that t r is non-orientable. Give aR and dR the respective orientations indicated in Figure 2.1.3c. By Basic Fact 2 of Proposition 1.3.7, there is a trainpath on t r which begins by traversing d'R and ends by traversing afR (in their specified orientations), and there is some sub-trainpath pR C Tq C t r with these same properties. Fix a branch f 0 of r f0 , and consider the branch f fL of t'l containing it; by recurrence of r£, there is some closed trainpath p on t !l traversing f'L . Identifying TO C T'R with Tq C in the natural

122

S E C T I O N 2.1

way, one produces a closed trainpath on t 0' traversing f'Q by replacing each appearance of e'L in p by the path pr C t '0 . Thus, f'0 lies in the maximal recurrent subtrack of Tq \ since f Q was arbitrary, recurrence of t 'q is assured, and the proof is complete under the assumption that r ’0 is connected. Finally, suppose that t 0* is disconnected and each of tr , r'L is recurrent. Let us concentrate on the component r+ of Tq containing a!Q. Since rR is recurrent and r'0 is disconnected, there is some trainpath pr on rR which is disjoint from e'R and begins and ends by traversing a'R with opposite orientations. Identify r'o C t 'r with Tq C t 'l as before, and suppose that f'0 C is a branch of Tq contained in the branch f L of t 'l . Since r'L is recurrent, there is some closed trainpath p on t 1ltraversing f'L , and since Tq is disconnected, p decomposes into the concatenation of trainpaths which either lie entirely in r+ or lie in tl — t+ and begin and end by traversing with opposite orientations. One then replaces each sub-trainpath of the latter type by a copy of the trainpath pR to produce a closed trainpath in 7+ traversing / 'Q . Since f o was arbitrary, r+ is recurrent, as desired. An analogous argument handles the component r 0f — r+ of t Q , ) and the proof is complete. q.e.d. R e m a rk : Some examples to show that that the latter possibilities in the previous lemma actually occur are given in Figure 2.1.4: in Figure 2.1.4a, only a collision along e yields a recurrent track, and in Figure 2.1.4b, only a left split along e yields a recurrent track.

F IG U R E 2.1.4 We say that two measured train tracks and (72,^ 2) are equiv­ alent if the associated positively measured tracks and (7*2^ , //2 ) are related by a composition of shifts, splits, collapses, and isotopies. We will usually regard isotopic tracks as identical. The equivalence class of a measured track (r, p) will be denoted [r, p]. If (r2, ^ 2)) where p 2 > 0> arises from ( ti,^ i) , where > 0, by a composition of shifts, splits, and isotopies (but no collapses), then we say that (ri,/ii) refines to ( t2,P2)- We may also say simply that t\ refines to r2 (with no mention of measures on the tracks) if there are measures /ii, p 2 as above. It follows from remarks above

SPLITTING, SHIFTING, A ND CARRYING

123

that if T\ refines to 7-2, then 7*2 < n , and, furthermore, T\ is recurrent if and only if 72 is recurrent. To close this section, we investigate the relationship between V^(r') and V ( t ) in case r f < r and let denote a supporting map for this carrying. We define the corresponding incidence matrix M ^ as follows. Enumerate the branches of r and r ; as {ft*}" and {6^}” , respectively. Let a* C F be an arc meeting r in a single point of 6,- which is a regular point of V v f v M ^

b

b" ^ (a )

~ \ ^

^ ( b)

(c )

F IG U R E 2.3.1 L em m a 2.3.2: For each trivalent switch v of the recurrent generic track r, there is a unique one-way trainpath on r starting at v. This trainpath is necessarily imbedded. P roof: Uniqueness of one-way trainpaths is clear, and we first concentrate 127

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on proving existence. There is a unique branch b of r which is large at v. If b is actually a large branch, then existence is trivial, and if b is only a half-large branch, then let vf denote the switch of r at which b is small. By recurrence of r, v and there is a unique branch 6' of r which is large at vf . If b7 is a large branch, then existence again follows easily, while if b' is small at the switch vn, then vu £ by recurrence, and there is a unique branch 6" £ {6,&'} which is large at vn. Continuing in this way, we produce ever-larger imbedded trainpaths whose first branch is large at v. Since r has only finitely many branches, this process terminates with an imbedded one-way trainpath starting from v, proving the lemma. q.e.d. Suppose that (r, p) is a measured generic track with p > 0 and v is a trivalent switch of r. We next define a sequence

of measured tracks where (rj+i,pj+i) arises from(Tj,pj) by a single split, for each j. To define this sequence, begin with j — 0, and consider the one­ way trainpath pj on Tj starting at v. If this trainpath has length one, so that there is a large branch incident on v, then the sequence of measured tracks terminates with (75,/ij); in the contrary case, perform a split on the last branch traversed by pj (which is a large branch) to produce ( tj+i , //j+i). Since pj is imbedded in Tj by the previous lemma, Tj+\ agrees with Tj in a neighborhood of v, and we let v also denote the corresponding switch of 1 . Continuing in this way, we produce the desired finite or semi-infinite sequence of measured tracks. We call this the process o f the triva lent switch v o f t with respect to the measure p . An example of such a process is given in Figure 2.3.1. In fact, this sequence of measured tracks is always finite, as we next show. P ro p o s itio n 2.3.3: I f v is a trivalent switch of the generic track t , then the process of v with respect to the measure p > 0 on r terminates. P ro o f: Since there can be at most a finite number of collisions during the process of a trivalent switch, we may assume without loss that there are, in fact, no collisions. Thus, each singular leaf in the interior of the bi-foliated neighborhood (N, J 7, T L ) associated to (r, p) in Step 1 of Construction 1.7.7 is semi-infinite and is equipped with a canonical orientation. We associate a complexity to the triple (r, p,v) as follows. Let q > 1 denote the length of the one-way trainpath on r which begins at r/, and define p > q to be the infimum of the lengths of the tie-transverse paths of (A, J 7, J7-1) corresponding to initial segments of singular leaves of T whose associated trainpaths terminate by traversing the half-branch of r which is

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large at v. This infimum must be finite since otherwise N would contain an imbedded region of infinite area (for the natural metric on N where each rectangle comprising N has length unity and width given by the measure p). Finally, the complexity of (r, p,v) is the ordered pair (p, q) with the lexicographic order. Now, consider the process of v with respect to p. There are two pos­ sibilities for a split from (Tj,pj) to (rj+1,/i;+1) (since we have assumed that there are no collisions) as in Figure 2.3.2a and b, which represent a right and left split, respectively. In the former case, p is either unchanged or decreased by one and q is decreased by one, while in the latter case, q may increase (subject to p > g, of course) but p must decrease by one. It follows that each split decreases the complexity, so the process of v must terminate, as was claimed. q.e.d.

(a)

(b) F IG U R E 2.3.2

R e m a rk : In effect, the previous proposition says that given a half-large branch b which is large at a switch v of a track r and given a measure p > 0 on r, we may split (r,p ) outside a neighborhood of v until the branch of the resulting track corresponding to b is large. P r o o f o f T h e o re m 2.3.1 : To begin, we claim that if (ri,p i) and (72 ,^ 2) are related by splitting and collapsing alone, then they split to a common measured track. Of course, if (r,-,/it-) arises from (73, /ij) by splitting alone, for { i ,j} = {1,2}, then the claim is trivial; similarly, if (77,p*-) splits to a measured track (rf,p ') which itself collapses to (73,/x/), for { i ,j} = {1,2}, then the claim follows by taking (r, p) = ( r 7,//) . The salient point for completing the proof of the claim is that collapses and splits commute in

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the following sense: if ( r ', / / ) arises from by a single collapse, for i = 1,2, then either (t* , pi) is isotopic to (r2, //2), or there is some measured track ( r ;/,p ;/) which collapses to both of (r,-,p,-), for £ = 1,2. Indeed, let 6,denote the branch of ( t ', / / ) along which one splits to produce (r,*,//,), for i = 1,2; if &i = 62, then (7*1 , p i) is isotopic to (r2)fi2), and if 61 ^ &2>fhen let (r'^ p " ) be the track obtained from ( t ', / / ) by performing splits along both of 61 and b2. It follows that the composition of a collapse followed by a split can be replaced by the composition of a split followed by a collapse, so any composition of collapses and splits can be put into one of the forms discussed above, proving the claim. Next, consider the case in which (71, pi) differs from ( t2,/ i 2) by a single shift and adopt the notation of Figure 2.3.3a, where the shift on r,is performed along a branch which is small at the switch v,- and large at the switch for i = 1,2. Let denote the complexity as defined in the proof of Proposition 2.3.3 above corresponding to the triple vj), for i = 1,2. Define a new lexicographically ordered “joint” complexity by 0 m ) = (Pi + JP2>9i +