princeton university press renormalization and 3-manifolds which fiber over the circle jul 1996

founda-Theorems and ConjecturesCompactness of'l/JnM Compactness of'R;f Thurston's double limit theorem Sullivan's a priori bounds Rigidity of double limits Rigidity of towers Compactness

Renormalization and 3-Manifolds which Fiber over the Circle

by

Curtis T McMullen

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY

1996

Copyright © 1996 by Princeton University Press

ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by

Luis A Caffarelli, John N Mather, and Elias M Stein

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

Printed in the United States ofAmerica by Princeton Academic Press

p cm - (Annals of mathematics studies: 142)

Includes bibliographical references and index.

ISBN 0-691-01154-0 (cl : alk paper) - ISBN 0-691-01153-2 (pb : alk paper)

1 Three-manifolds (Topology) 2 Differentiable dynamical systems.

I Title II Series: Annals of mathematics studies: no 142.

QA613 M42 1996

514'.3-dc20 96-19081

The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed

Structures on surfaces and 3-manifolds

Quasifuchsian groups

The mapping class group

Hyperbolic structures on mapping tori

Asymptotic geometry

Speed of algebraic convergence

Example: torus bundles

Contents

1 Introduction

2 Rigidity of hyperbolic manifolds

2.1 The Hausdorff topology

2.2 Manifolds and geometric limits

4.2 Polynomials and polynomial-like maps

4.4 Improving polynomial-like maps

4.5 Fixed points of quadratic maps

4.6 Renormalization

4.7 Simple renormalization

4.8 Infinite renormalization

5 Towers

5.1 Definition and basic properties

5.2 Infinitely renormalizable towers

5.3 BOllIH)pd (,()luhinatorics

5.4 rtohIlHf.IU'HH and illlH'r rip;idity

111

· 11

121724323441 42

4446505360657575

· 76

7883868891

92

95

· 95

9799 101

· 106 108 112

· 114

· 117

7.1 Framework for the construction of fixed points 1197.2 Convergence of renormalization 1277.3 Analytic continuation of the fixed point 1287.4 Real quadratic mappings 132

8 Asymptotic structure in the Julia set

8.1 Rigidity and the postcritical Cantor set

8.2 Deep points of Julia sets

8.3 Small Julia sets everywhere

9.4 Quadratic maps and universality

9.5 Speed of convergence of renormalization

10 Conclusion

Appendix A Quasiconformal maps and flows

A.l Conformal structures on vector spaces

A.3 BMO and Zygmund class

A.4 Compactness and modulus of continuity

· 183

· 186 191 196 200

CONTENTS vii

205 .205

· 217

· 221

· 232.234

.237

Appendix B Visual extension

B.l Naturality, continuity and quasiconformality

B.2 Representation theory

B.3 The visual distortion

B.4 Extending quasiconformal isotopies

Renormalization and 3-Manifolds which

Fiber over the Circle

1 Introduction

In the late 1970s Thurston constructed hyperbolic metrics onInost 3-manifolds which fiber over the circle Around the same time,Feigenbaum discovered universal properties of period doubling, andoffered an explanation in terms of renormalization Recently Sulli-van established the convergence of renormalization for real quadraticInappings

In this work we present a parallel approach to renormalizationand to the geometrization of 3-manifolds which fiber over the cir-cle This analogy extends the dictionary between rational maps andKleinian groups; some of the new entries are included in Table 1.1

DictionaryKleinian group f ~ 1Tl(8) Quadratic-like mapf :U -+V

Mapping class1/J :8 ~ S Kneading permutation

'l/J :AH(S) -+ AH(S) Renormalization operator R p

Cusps in 8By Parabolic bifurcations in M

Totally degenerate Infinitely renormalizable

group r polynomial f(z) = z2+c

Ending lamination Tuning invariant

Fixed point of'ljJ Fixed point ofRp

Hyperbolic structure on Solution to Cvitanovic-Feigenbaum

M3 -+81 equation fP(z) =a.-I f(a.z)

Table 1.1

Both discussions revolve around the construction of a nonlineardynamicalSystClll which is conformally self-similar

for some a E Aut(C) Such a p is a fixed point for the action of

1/1 on conjugacy classes of representations This fixed point gives ahyperbolic structure on the 3-manifold

Tt/J = S x [0,l]/(s,0) f'.J ('l/J(s), 1)

which fibers over the circle with monodromy 1/J. Indeed, the mal automorphisms of the sphere prolong to isometries of hyperbolicspace lH[3, and T"p is homeomorphic to JH[3/r where r is the groupgenerated by a and the image ofp.

confor-For renormalization the sought-after dynamical system is a degreetwo holomorphic branched covering F : U ~ V between disks U C

V c C, satisfying the Cvitanovic-Feigenbaum functional equation

for some a E C* The renormalization operator 'R p replaces F by

its pth iterate FP, suitably restricted and rescaled, and F is a fixed

point of this operator

In many families of dynamical systems, such as the quadraticpolynomials z2 +c, one sees cascades of bifurcations converging to

a map f(z) =z2+COO with the same combinatorics as a fixed point

of renormalization In Chapter 9 we will show that R,;(f) converges

exponentially fast to the fixed point F Because of this convergence,

quantitative features ofF are reflected in f, and are therefore versal among all mappings with the same topology

uni-Harmonic analysis on hyperbolic 3-space plays a central role indemonstrating the attracting behavior of ~p and 'l/J at their fixedpoints, and more generally yields inflexibility results for hyperbolic

3-manifolds and holomorphic dynamical systems

INTRODUCTION 3

We now turn to a more detailed s~mmary.

Hyperbolic manifolds By Mostow rigidity, a closed hyperbolic:J-manifold is determined up to isometry by its homotopy type Anopen manifold M = ]8[3/r with injectivity radius bounded aboveand below in the convex core can generally be deformed However,such an M is naturally bounded by a surface 8M with a conformalstructure, and the shape of8M determinesM up to isometry

In Chapter 2 we show these open manifolds with injectivity boundswhile not rigid, are inflexible: a change in the conformal structure

0118Mhas an exponentially small effect on the geometry deep in theconvex core (§2.4) This inflexibility is also manifest on the sphere atinfinity it: a quasiconformal conjugacy from r = 7rl(M) to anotherKleinian group r' is differentiable at certain points in the limit set

A These deep points x E Ahave the property that the limit set isnearly dense in small balls about x - more precisely, Acomes withdistancer1+e of every point in B (x, r).

Chapter 3 presents a variant of Thurston's construction of perbolic 3-manifolds that fiber over the circle Let 1/J :S ~ S be apseudo-Anosov homeomorphism of a closed surface of genus 9 ~ 2.Then the mapping torus T1/J is hyperbolic To construct the hyper-bolic metric onT1/J' we use a two-step iterative process

hy-First, pick a pair of Riemann surfaces X andY in the Teichmiillerspace of S. Construct the sequence of quasifuchsian manifolds

Q ('ljJ-n (X) , Y), ranging in a Bers slice of the representation space

of 7rl(S) Let M = limQ('ljJ-n(x) , Y). The Kleinian group senting 7rl(M) is totally degenerate - its limit set is a dendrite.For the second step, iterate the action of1/J on the space of rep-resentations of 7rl(S), starting with M. The manifolds 'ljJn(M) areall isometric; they differ only in the choice of isomorphism between

repre-1rl(M) and1rl(S). A fundamental result of Thurston's - the doublelimit theorem - provides an algebraically convergent subsequence

(t/)n(M) ~ M"po The theory of pleated surfaces gives an upper bound

on the injectivity radius ofM in its convex core Therefore any 'rnetriclimit N of1/Jn(M) is rigid, and so'ljJ is realized by an isometry

geo-l.~ : M1/J ~M¢, completing the construction

Mostow rigidity implies the full sequence converges to M1j;. Fromthe the inflexibility theory of Chapter2, we obtain the sharper state-

1l1entthat 'ljJn(AiJ) ~ A1,/I exponentially fast

4 CHAPTER 1 INTRODUCTION

The case of torus orbifold bundles over the circle, previously sidered by J(2jrgensen, is discussed in §3.7 We also give an explicitexample of a totally degenerate group with no cusps (see Figure 3.4).Renormalization The simplest dynamical systems with criticalpoints are the quadratic polynomials I(z) = z2 +c In contrast

con-to Kleinian groups, the consideration of limits quickly leads one con-tomappings not defined on the whole sphere

A quadratic-like map 9 : U ~ V is a proper degree two morphic map between disks in the complex plane, withU a compact

holo-subset of V Its filled Julia set is K(g) =ng-n(v).

If the restriction of an iterate In :Un ~Vn to a neighborhood of

the critical point z = 0 is quadratic-like with connected filled Juliaset, the mappingIn is renormalizable When infinitely many such n

exist, we sayI is infinitely renormalizable.

Basic results on quadratic-like maps and renormalization are

pre-sented in Chapter 4 In Chapter 5 we define towers of quadratic like

maps, to capture geometric limits of renormalization A tower

T =(Is : Us ~ Vs : s E S)

is a collection of quadratic-like maps with connected Julia sets,

in-dexed by levels s > o. We require that 1 E S, and that for any

s,t E S with s < t, the ratio t/s is an integer and fs is a

renormal-ization of/:/s.

A tower has bounded combinatorics and definite moduli ift/s isbounded for adjacent levels and the annuli V, - Us are uniformlythick In Chapter 6 we prove the Tower Rigidity Theorem: a bi-infinite tower T with bounded combinatorics and definite moduliadmits no quasiconformal deformations (This result is a dynamicalanalogue of the rigidity of totally degenerate groups.)

To put this rigidity in perspective, note that a single like map 11 :Ul ~VI is never rigid; an invariant complex structurefor11 can be specified at will in the fundamental domainVI-Ul. In atower with infS =0, on the other hand, 11 is embedded deep withinthe dynamics ofIs for s near zero (since li/s= 11) The rigidity oftowers m~kes precise the intuition that a high renormalization of aquadratic-like map should be nearly canonical

quadratic-Chapter 7 presents a two-step process to construct fixed points ofrenormalization The procedure is analogolls to that used to find a

[NTRODUCTION 5

fixed-point of1/; For renormalization, the initial data is a real number

csuch that the critical point ofz ~ z2+c is periodic with periodp.

'fhe first step is to construct an infinite sequence of superstable points

c*nin the Mandelbrot set by iterating the tuning map x ~c*x The

iterated tuningsc· n converge to a pointCOO such that f(z) = z2+COO

is infinitely renormalizable In the classical Feigenbaum example,(~ == -1 and c· n gives the cascade of period doublings converging tothe Feigenbaum polynomialf(z) == z2+(-1)00 =z2-1.4101155···.rrhe maps f and Rp(f) are quasiconformally conjugate near theirJulia sets

The second step is to iterate the renormalization operator R p ,

starting with the pointf. By Sullivan's a priori bounds, we can pass

to a subsequence such that'R;(f) converges to a quadratic-like map

F. This F can be embedded in a tower

T == (fs : s E S == { ,p-2,p-l, l,p,p2, }),

such that fl == F and fpk = lim'R;+k(f). In the limit we also have

a quasiconformal mapping <p :C -+C conjugating fps to fs for each

s. By the Tower Rigidity Theorem, <p is a conformal map (in fact

¢(z) =az), and thus F is a fixed point of renormalization Just asfor 3-manifolds, we use rigidity of the geometric limit T to conclude

the dynamics is self-similar

Much of the construction also works when c is complex, but somesteps at present require creal

Deep points and uniform twisting Chapters 8 and 9 developresults leading to the proof that renormalization converges exponen-tially fast

Chapter 8 exploits the tower theory further to study the etry and dynamics of infinitely renormalizable maps f(z) == z2 +c.Assumef has bounded combinatorics and definite moduli Then weshow:

geom-1 The complement of the postcritical Cantor set P(f) is a mann surface with bounded geometry

Rie-2 The critical pointz =0 is a deep point of the Julia setJ(f). Inparticular, blowups ofJ(f) about z== 0 converge to the wholeplane in the Hausdorff topology

holomor-Roughly speaking, (F,A) is uniformly twisting if the geometriclimits ofF as seen from within the convex hull of A inlH[3 are verynonlinear Condition (3) above implies that (:F(/), J(/)) is uniformlytwisting, whereF(f) is the full dynamics generated by f and 1-1.

The Deep Conformality Theorem asserts that a quasiconformalconjugacy between uniformly twisting systems is C1+O:-conformal atthe deep points of A By (2) above, we conclude that any conjugacyfrom I to another quadratic-like map is differentiable at the criticalpoint z= O Exponentially fast convergence of renormalization thenfollows

To state the final result, we remark that a fixed point of izationF has a canonical maximal analytic continuationF :W ~c.

renormal-The domainW ofFis an open, dense subset of the plane Then there

is a A<1 such that for any compact K c W,

for all I with the same inner class as F, and alln » O

The fixed point constructions for mapping classes and for malization operators are compared retrospectively in Chapter 10

renor-We conclude with some open problems; among them, the conjecturalself-similarity of the boundary of Teichmiiller space, as observed incomputer experiments conducted jointly with Dave Wright Theseparallels and open questions are summarized in Table 1.2

Harmonic analysis Two appendices develop the analytic tions of our results Appendix A is devoted to quasiconformal flows

founda-Theorems and Conjectures

Compactness of'l/Jn(M) Compactness of'R;(f)

(Thurston's double limit theorem) (Sullivan's a priori bounds)

Rigidity of double limits Rigidity of towers

Compactness + rigidity Compactness + rigidity

Exponential convergence Exponential convergence

Injectivity radius bounded Quadratic-like dynamics atabove in the convex core every scale

A is locally connected J is locally connected

Geodesic flow ergodic Tower dynamics ergodic?Bers' boundary self-similar? Mandelbrot set self-similar?Ending lamination conjecture Mandelbrot set locally connected?

Table 1.2

8 CHAPTER 1 INTRODUCTION

and Reimann's theorem, that a vector field with strain in Loo erates a unique quasiconformal isotopy The proof is streamlined byshowing a function with first derivatives in BMO satisfies a Zygmundcondition

gen-Appendix B is devoted to the visual extension of deformationsfrom S~-l to lHIn

. This extension has been studied by Ahlfors,Reimann, Thurston and others

A key role for us is played by the visual distortion M v(p) of a

vector field v on S~-l, as seen from p E IBm By definition Mv(p)

is the minimum, over all conformal vector fields w, of the maximum

visual length of(v-w)as seenfromp. The quantityMvdepends only

on the strainSv (the Beltrami differential J-t=8vwhen n= 3), but

in a subtle way Our inflexibility results are all proved by bounding

Mv.

An illustration The intuitive link between rigidity, deep points

in the Julia set and convergence of renormalization is the following.Let f be the Feigenbaum polynomial, and let ¢ be a quasiconformal

conjugacy between f and f 0 f. The dilatation of c/> is specified

by a field of infinitesimal ellipses supported outside the Julia set

J(f) and invariant under the action of f. To visualize a typicalinvariant ellipse field, first consider a family of ellipses in the plane

of constant eccentricity whose major axes are along rays through theorigin This ellipse field is invariant under the mapping z ~ z2.

Now the Riemann mapping from the outside of the unit disk to theoutside of the Feigenbaum Julia set transports the dynamics of z2

to that off. Since the Julia set is quite dense near the postcriticalset, the argument of the derivative of the Riemann mapping varieswildly, and so the resulting ellipse field is more or less random Thestretching in different directions approximately cancels out, so that

¢, while fluctuating at very small scales, is close to a conformal mapnear the critical point The Julia set of a high renormalization off

also resides in a small neighborhood of the critical point Since¢also

conjugates f2 n to f2 n + 1 , these two mappings are nearly conformallyconjugate, and in the limit we obtain a fixed point of renormalization

A blo~up of the Feigenbaum Julia set near the critical pointappears in Figure 1.3 The tree-like black regions are the pointsoutside the Julia set The thin postcritical Cantor set, lying on thereal axis and evidently well-shielded from the ellipse ficIcI, is also

Figure 1.3 Asymptotic rigidity near the postcritical set.

struc-Feigenbaum's work appears in [Feig]; similar discoveries weremade independently by Ooullet and Tresser [OoTr] Manyaq.ditionalcontributions to the theory of renormalization are collected in [Ovi] ·

Sullivan's a priori bounds and a proof of convergence of

renormaliza-tion using Riemann surface laminarenormaliza-tions appear in [Su15] and [MeSt].The dictionary between rational maps and Kleinian groups wasintroduced in [SuI4] See [Mc5] for an illustrated account, centering

on classification problems for conformal dynamical systems

A brief discussion of the results presented here and other tions on Teichmiiller space appeared in [Mc3] This work is a sequel

itera-to [Mc4], which discussed the foundations of renormalization, and itera-to[Mc2] , which began the program of using analytic estimates in an

"effective" deformation the~ryfor Kleinian groups

Parts of this work were presented at CUNY and IHES in 1989and 1990, at the Boston University Geometry Institute in 1991 and inthe Keeler Lectures at University of Michigan in 1993 A preliminaryversion of this manuscriptwascirculated in Fall of 1994 Many usefulcorrections and suggestions were provided by G Anderson and thereferee This research was funded in part by the Miller Institutefor Basic Research and the NSF I would like to thank all for theirsupport

2 Rigidity of hyperbolic lllanifoids

This chapter begins with basic facts about complete hyperbolic

I ••a.nifolds and their geometric limits We then give a proof of rigidityfor manifolds whose injectivity radius is bounded above Mostowrigidity for closed manifolds is a special case; the more general resultwillbe used in the construction of hyperbolic manifolds which fiber()ver the circle

The proof of rigidity combines geometric limits with the Lebesgue(tensity theorem and the a.e differentiability of quasiconformal map-pings This well-known argument is carried further in §2.4 to showcertain open hyperbolic manifolds, while not rigid, are inflexible -

any deformation is asymptotically isometric in the convex core Thisinflexibility is also manifest on the sphere at infinity: quasiconformal

•

conjugacies are automatically differentiable and conformal at certain

I>oints in the limit set These results will be applied to surface groups

in Chapter3

Useful references on hyperbolic geometry include [CEG] , [BPJ,

['fhl] and [Th4]

We will frequently need to make precise the idea that certainluanifolds, dynamical systems, compact sets or other objects con-vergegeometrically To formulate this notion, we recall the Hausdorff

12 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

Given an arbitrary sequence (Fi) of closed sets in X, we define

lim infFi as the largest set satisfying condition (a), and lim sup Fi

as the smallest set satisfying condition (b) Both lim infFi and

lim supFi are closed and lim inf Fi C lim supFi We have F i ~ F if

and only if lim infFi = lim supF i = F.

Proposition 2.1 The space CI(X) is sequentially compact in the Hausdorff topology.

Proof [Haus, §28.2]: Let F i be a sequence in CI(X), and let Uk be

a countable base for the topology on X For each k, ifUk nFi :f 0for infinitely many i, then we may pass to a subsequence such that

Uk nFi =I 0for all but finitely many i Diagonalizing, we obtain asubsequence Fin which converges Indeed, if x E lim supFin' then

for any neighborhoodU of X, Fin nU=I 0for infinitely manyn But

thenFin meets U for all but finitely many n, and thus x E lim infFin.

Since the upper and lower limits agree, the sequence converges

•

Now suppose X is also locally compact By separability, X can

be exhausted by a countable sequence of compact sets, so its point compactificationX* =Xu {oo} is metrizable For each closedsetF c X, let F* = F U{oo} c X*, and define

one-6(F 1 , F 2 ) = inf{€ > 0 : F; is contained in an €-neighborhood of

References: [Haus], [HY, §2-16]' [Nad)

Definitions Hyperbolic space IBm is acomplete simply-connectedn-manifold of constant curvature -1; it is unique IIp to isometry

2.2 MANIFOLDS AND GEOMETRIC LIMITS 13

'fhe Poincare ball gives a model for hyperbolic space as the unit ball

in IRn with the metric

d 2 _ 4dx 2

S - (1 _ r 2)2·

'fhe boundary of the Poincare ball models the sphere at infinityS~-l

for hyperbolic space, and the isometries of IHrn prolong to conformalluaps on the boundary

In dimension three, the sphere at infinity can be identified witht.he Riemann sphereC,providing an isomorphism between the orien-t.ation preserving group Isom+(IHrn) and the group of fractional lineartransformations Aut(C) ~ PSL 2 (C).

A Kleinian groupr is a discrete subgroup of Isom(IHrn) A Kleinian

~roupis elementaryif it contains an abelian subgroup of finite index

A hyperbolic n-manifold M is a complete Riemannian manifold

of constant curvature -1 Any such manifold can be presented as aquotient M = IHrnjr of hyperbolic space by a Kleinian group

Orientation All hyperbolic manifolds we will consider, including

IHrn itself, will be assumed oriented The identification between ]H[n

and the universal cover of M will be chosen to preserve orientation.

Then the group r = 1rl(M) is contained in Isom+(JH[n) and it is(ietermined byM up to conjugacy

The thick-thin decomposition The injectivity radius ofa

hy-perbolic manifold M at a point x is half the length of the shortest

essential loop through x.

The Margulis Lemma asserts that a discrete subgroup oflsom(IHrn)generated by elements sufficiently close to the identity contains anabelian subgroup of finite index [BP, §D], [Th4, §4.1] This result

controls the geometry of the thin part M(O,e] of a hyperbolic fold, Le the subset where the injectivity radius is less than f. There

mani-is an €n > 0 such that every component L of M(O,en] is either a

col-lar neighborhood of a short geodesic, or a cusp, homeomorphic to

N x [0,(0) for some complete Euclidean (n - I)-manifold In theuniversal coverlHI n

, each component L of the thin part is covered byeither an r-neighborhood of a geodesic, or by a horoball

The limit set A c ~c.;(r.:u-l of a Kleinian group,.r is the set of

ac-clIlllulation point.s of I';r for allY x E lHITI.; it, is indepcndpIlt of x.

14 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

For E C S~-l, the convex hull of E (denoted hull(E)) is thesmallest convex subset of lHIn containing all geodesics with both end-points inE. The convex core K of a hyperbolic manifoldM ==lHInIr

is given by K =hull(A)/f The convex core supports the recurrentpart of the geodesic flow; it can also be defined as the closure of theset of closed geodesics We say M is geometrically finite if a unitneighborhood of its convex core has finite volume

The open manifold M can be prolonged to a Kleinian manifold

wheren= S~-l-Ais the domain of discontinuityoff. In dimension

n == 3, ncan be identified with a domain on the Riemann sphere onwhich r acts holomorphically, so

8M=n/rcarries the structure of a complex one-manifold (possibly discon-nected)

To pin r down precisely, one may choose a frame w over a point

p EM; then there isituniquer such that the standard frame at theorigin in the Poincare ball lies over the chosen framew on M. Con-versely, any discrete torsion-free group r C Isom+(lHIn) determines

a manifold with baseframe (M, w) by taking M == lHIn/f and W =

the image of the standard frame at the origin When we speak ofproperties ofM holding atthe baseframew, we mean such properties

hold at the point pover which the baseframe lies

Geometric limits The geometric topologyon the space of bolic manifolds with baseframes is defined by (M i , Wi) +- (M, w) ifthe corresponding Kleinian groups converge in the Hausdorff topol-ogy on closed subsets of Isom(lHF) In this topology, the space ofall hyperbolic manifolds (M, w) with injectivity radius greater than

hyper-r>0 at the baseframew is compact.

Here is a more intrinsic description of geometric convergence:

(Mi,Wi) ~. (M,w) if and only if, for each compact submanifold

K c M containing the baseframe w, ~here are smooth embeddings

Ii : K + Mi' defined for all i sufficiently large, such that Ii sends

w to Wi and Ii tends to an isornetry in the Coo topolop;y rrhe last

2.2 MANIFOLDS AND GEOMETRIC LIMITS 15

eondition can be made precise by passing to the universal cover: then

we obtain mappings h :K -+ IBm, sending the standard baseframe

at the origin to itself; and we require that fi tends to the identityInapping in the topology of Coo-convergence on compact subsets of

to the boundary of the convex core of Mi tends to infinity.

Then the corresponding limit sets Ai converge to S~-l in the Hausdorff topology.

Proof Let Bi C S~-l - Ai be a spherical ball of maximum radiusavoiding the limit set The circle bounding Bi extends to a hyper-plane Hi in IBm bounding a half-space outside the convex hull of thelimit set Since the origin of the Poincare ball corresponds to thebaseframeWi, the hyperbolic distance from the origin to Hi is tend-ing to infinity But this means the spherical radius of B i is tending

to zero, so for i large the limit set comes close to every point on thesphere

•

If ri -+ r is a geometrically convergent sequence of Kleinian

~roups, then

A(r) c liminfA(ri)'

This follows from that fact that repelling fixed points of elements

of r are dense in its limit set However, the limit set can definitelyshrink in the limit For example, the Fuchsian groups

f(p) = {I' E PSL2(Z) : I'==Imodp}

converge geometrically to the trivial group (with empty limit set) as

p~ 00, even thoughA(r(p)) =S~ for allp (sinceIHI2jr(p) has finitevolume)

The situation is more controlled if the injectivity radius is bounded

above. Given R >r > 0, let 1t~R denote the space of all hyperbolic

16 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

1 the baseframe w is in the convex core of M;

2 the injectivity radius ofM is greater thanr at W; and

3 the injectivity radius is bounded above by R throughout theconvex core ofM.

Proposition 2.4 The space'H~R is compact in the geometric ogy, and the limit set varies continuously on this space.

topol-Proof First let M = IHrnIf be any hyperbolic manifold In terms

of the universal cover, the injectivity radius at a point x is given by

r(x,r) = -2

1inf d(x, ')'x).

Indeed, a point y E M where the injectivity radius is less than R lies

on an essential loop of length at most2R; shrinking this loop, we find

y lies within a distance D (depending only on R) of either a closedgeodesic or a component of the Margulis thin part ofM. Lifts of theclosed geodesic lie in the convex hull of the limit set; lifts of the thinpart touch the sphere at infinity in the limit set and have Euclideandiameters tending to zero In either case, we conclude that a point

y E T(r, R) which is close to the sphere at infinity is also close tothe limit set, and the claim follows

Now let (Mi, Wi) be a sequence in 'H~R; by the lower bound onthe injectivity radius, we can assume the sequence converges geo-metrically to some based manifold (M,w). Let f i ~ f be the corre-sponding sequence of Kleinian groups

For any D and x E IHrn, the set of hyperbolic isometries with

d(x, IX) :5 D.is compact, so we have r(x,fi) ~r(x,f) uniformly oncompact subsets oflHIn

. ThereforelimsupT(r i , R) c T(r, R)

2.3 RIGIDITY 17

with respect to the Hausdorff topology on closed subsets ofIHrn Byhypothesis, the injectivity radius is bounded above by R in the con-vex core ofM i , so hull(Ai) is contained inT(ri, R). Therefore

limsuphull(Ai)C T(r,R).

Since the origin lies in hull(Ai ) for all i, T(r, R) contains all limits

()frays from the origin to Ai, and thus

Definitions A diffeomorphism f :X -+ Y between Riemannian'n-manifolds is an L-quasi-isometryif

.! < IDf(v)1 < L

-for every nonzero tangent vectorv to X.

A homeomorphism ¢ :X + Y (for n > 1) is K -quasiconformal

if<phas distributional first derivatives locally in Ln, and

for almost every x and every nonzero vectorv E TxX (see §A.2)

A hyperbolic manifold M = IHrnIr is quasiconformally rigid if

anyquasiconfornlal nutp 4) :""":~l t -+ A.,,~-l, conjugating r to another

18 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

Kleinian group r', is conformal Similarly M is quasi-isometrically rigidif any quasi-isometryf :M ~ M', where M' is also hyperbolic,

is homotopic to an isometry

The following result is well-known

Theorem 2.5 Letf :Ml + M2 be a ~-quasi-isometry between perbolic n-manifolds Mi = w /ri, i = 1,2 Then the lift

hy-I:JHr~IHr

ofI to the universal covers extends continuously to a K(~)-quasi­

conformal map on S~-l conjugating rl to r2. The constant K(~)

The quasiconformality of the boundary values of quasi-isometries

is a key step in the proof of Mostow rigidity, and is true under weakerhypotheses(Ineed only distort large distances by a bounded factor);see [Mos], [Thl, §5] For the converse in dimension three, see Corol-lary B.23

Corollary 2.6 IIM is a quasiconformally rigid hyperbolic n-manifold, then it is quasi-isometrically rigid.

Proof Given a quasi-isometry f :M ~ M', the lifted map! :

W ~ W extends to a quasiconformal conjugacy between rand

r'; by hypothesis the boundary mapping is actually conformal, so it

agrees with the boundary values of an equivariant isometryaof W

A homotopy can be constructed by interpolating along the geodesicjoining!(x) to a(x) for each x.

•

2.:L RIGIDITY 19

Ilivariant line fields We now identify the sphere at infinity S~

withthe Riemann sphereC. Let L 1(C,dz 2 ) denote the Banach space

of measurable integrable quadratic differentials 'ljJ='ljJ(z)dz 2 on theHphere, with the norm

'(lhe absolute value ofa quadratic differential is an area form, so the

Ilorm above is conformally natural; that is, for any A E Aut(C),

IIA*4>11 = 114>1/·

The dual of L 1(C,dz 2 ) is M(C) = LOO(C, d:Z/dz) , the BanachHI)a.ce of bounded measurable Beltrami differentials J.L(z) az/dz

C'(luipped with the sup-norm The pairing between L 1(C,dz 2 ) and

I/X>(C, az/dz) is given by

it is also conformally natural

The weak* topology on M(C) is defined by J.Ln ~ J.L if and only if

for every 'ljJ E L 1(C,dz 2 ).

A line field is a Beltrami differential with IJ.LI = 1 on a set E of

positive measure and IJ.LI =0 elsewhere The tangent vectorsesucht.ha.t /-L(e) = 1 span a measurable field of tangent lines over E, and

Hitch a line field determines J.L.

l)roposition 2.7 A hyperbolic 3-manifold M = lH[3/r is ltn."nally rigid if and only if there is nor-invariant measurable line jit'ld on the sphere at infinity.

quasicon-Ilroof The complex dilatationJ.L = 4>z/<Pz of a quasiconformal jup;acy between a pair of Kleinian groups rand r' is a r-invariant

con-I~eltrami differential; if 4> is not conformal, then J.L/IJ.LI provides anI'-invariant line field on the set of positive measure where J.L # o.(~()l1vcrsely, ifJ.L is an invariantline field, then for any complext with//.I < 1 thereis a quasiconfornHl.lluappiull:cPt with dilatation tJL [AB],

20 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

and 4Jtconjugatesr to another Kleinian groupr'. (Conceptually, ttL

is a new complex structure invariant underr; cPtprovides a change ofcoordinates transportingttLto the standard structure on the sphere.)

•The unit ball in M(C) is compact in the weak* topology Thusany sequence of line fields tLn has a weak*-convergent subsequence.However, the weak* limit need not be a line field: for example, thelimit tL may equal 0 if /In is highly oscillatory In any case we stillhave:

Proposition 2.8 Ifrn ~ r geometrically and tLn are invariant line fields for rn, then any weak* limit tL of tLn isr-invariant.

Proof For any 'l/J E L 1(C,dz 2 ), , E rand f > 0, there is a borhood U of , in Aut(C) such that 116'.1P - ,.1P/I <f for all 6' in U.

neigh-By the definition of geometric convergence, there is a "'In ErnnU

for all n sufficiently large, and thus:

I(tLn - "'I JLn, 'l/J) I = I(/In, 'l/J - "'I.'l/J)I <

A groupr is quasiconformally rigid on its limit set if there is no

r-invariant line field supported onA This means any quasiconformalconjugacy4>which is conformal outsideAisaMobius transformation

A line field is parabolic if it is given by JL = A*(dZ/ dz) for some

A E Aut(C) A parabolic line field is tangent to the pencil of circlespassing through a given point in a given direction; see Figure 2.1.When/J=.dZ/ dz, the circles become the horizontal lines in the plane

We now show that any hyperbolic manifold whose injectivity dius is bounded above is quasiconformally rigid More generally wehave:

ra-~.3 RIGIDITY

Figure 2.1 A parabolic line field on the sphere

21

'fheorem 2.9 (Bounded rigidity) A hyperbolic 3-manifold M =

IIn:l/r whose injectivity radius is bounded above throughout its convex (tor"e is quasiconformally rigid on its limit set.

Ilroof Suppose to the contrary that r admits an invariant line field

I'· on its limit set By the Lebesgue density theorem, there is a point

I' E C where 1J.t(p)I =1 and J.t is almost continuous; that is, for any

Orient 9 towards z = 0, and consider its image in M. Either 9

rt'tllrns infinitely often to the thick part of M, or 9 enters a

com-ponent of the thin purt of 1\:1 nlld never exits In the latter case,

22 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

Z = 0 must be the fixed point of some nontrivial element of r (ahyperbolic element for a short geodesic, or a parabolic element for

a cusp) Since r is countable, we can choose p so these cases areavoided, and therefore 9 recurs infinitely often to the thick part of

leaves invariant the line field IJ,n = A~(IJ,), and IJ,n converges weak*

to the parabolic line field J.L00 = J.L(O)dZ/dz as n tends to infinity.

By construction, rn belongs to 1i~R where R is an upper bound

on the injectivity radius in the convex' core ofM. Thus we can pass

to a subsequence such that rn ~ r00 E 1-l~R. The limiting group

r00 leaves the parabolic line field Iloo invaria~t

Since z =0 is a point of Lebesgue density of the limit set A(r),

the magnified limit sets

converge to the whole sphere in the Hausdorff topology Thus thelimit set ofroo is also the whole sphere (by Proposition 2.4)

But any automorphism of the sphere preservingIloo must fix thepoint at infinity, contradicting the fact that every orbit ofr00 on thesphere is dense

hy-an ellipsold in the thy-angent space to almost every point The tors maximizing the ratio (¢*cr) (v) / 0"( v) span a canonical subspace

vec-Ex C TxS~-l, which cuts the ellipsoid in a round sphere of

max-imum radius (On the 2-sphere, Ex is just the line field of major

2.3 RIGIDITY 23

a.xes of the ellipses, or the whole tangent space at points where Dtj>

is conformal.)

If D</J is not conformal a.e on the limit set, then there is a set

of positive measure F c A over which rankEx is a constant k with

o < k < n - 1 Then ExlF is a r-invariant k-plane field Given

a.n upper bound on the injectivity radius of the convex core of M =

1I1In/r, we can blow up a point of almost continuity ofExIF, pass to

a geometric limit and obtain a contradiction as in the case of linefields In summary we have:

Theorem 2.10 Let M =lHr/r be a hyperbolic manifold of sion n ~ 3 whose injectivity radius is bounded above throughout its convex core Then r admits no measurable invariant k-plane field

dimen-an its limit set, 0< k <n - 1, and M is quasiconformally rigid.

Note that r need not be finitely generated

Ergodicity versus rigidity By a more subtle argument, SullivanHhows a Kleinian group admits no invariant k-plane field (0 < k <

'n-l) on the partofS~-lwhere its actionisconservative[Su13] (Thea.ction of a discrete group on a measure space is conservative if there is

nosetA of positive measure such that the translates{1'(A) : l' E r}a.re disjoint.) It is easy to show that an upper bound on the injectivityradius in the convex core implies r acts conservatively on its limitset Thus Sullivan's result implies the preceding Theorem

On the other hand, there exist hyperbolic 3-manifoldsM = lH[3/r

with bounded injectivity radius such that r does not act ergodically

on the sphere Thus ergodicity is stronger than rigidity

For example, let M be the covering space of a closed hyperbolic:l-manifold N induced by a surjective mapping 7T'1(N) ~ Z*Z (see

I~'igure 2.2) The injectivity radius of M is bounded above since

the kernel of p is nontrivial (a closed manifold does not have theIlomotopy type of a bouquet of circles) Almost every geodesic onMwanders off to one of the ends of the free group, which form a CantorHet Those landing in a given nonempty open set of ends have positiveIlleasure, and determine a r-invariant set of positive measure on thespllere Since a Cantor set is the union of two disjoint nonemptyopensets, the sphere is the union of twor-invariant sets of positiveInCaSllre

M

N

CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

Figure 2.2 A Z*Z covering space

The origins of the line field viewpoint can be found in [Ah5] and[SuI3]

2.4 Geometric inflexibility

Let M be a hyperbolic 3-manifold whose injectivity radius is

bounded above and below in its convex core, but with8M # 0. Then

M need not be rigid; deformations are often possible by changing theconformal structure on8M.

In this section we push the logic of geometric limits further to

show a deformation of M decays exponentially fast within its convex core In other words, the geometry of M deep within the core is

inflexible: it changes only a small amount, even under a substantialdeformation of8M. Our main result is:

Theorem 2.11 (Geometric inflexibility) Let \11 : M ~ M' be an L-quasi-isometry between a pair of hyperbolic 3-manifolds Suppose the injectivity radius of M in its convex core K ranges in the interval

[Ro,Rl], where Ro >0

Then there is a volume-preserving quasi-isometry ~ : M ~ M', boundedly homotopic to \11, such that the pointwise quasi-isometry constant L(~,p) satisfies

L(4.>,p) ~ 1+Cexp(-ad(p,M - K)).

WhenK = M, the Theorem says\11 is homotopic to an isometry.,rhus Mostow rigidity for closed manifolds is a special case, and the,I'heorem can be thought of as an "effective" version of rigidity foropen manifolds These effective bounds are most interesting when

M is geometrically infinite - then iI> is exponentially close to aniHometry deep in the convex core But the Theorem also has contentwhen the convex core is compact, because the constants depend on

M only via its injectivity radii

Figure 2.3 The visual extension from 8M to M.

Idea of the proof A vector field v on S~-l has a canonicalvisual

(':t:tension to a vector field V =ex(v) on JHrn When v represents a(IUasiconformal deformation of8M, V gives a quasi-isometric defor-Illation ofM. Forp E M,the metric distortionSV(p) is the expectedvalue of the quasiconformal distortion Sv at the endpoint of a ran-dom geodesic ray, from p to 8M (see Figure 2.3) Because of theilljectivity bounds on M, a geodesic starting deep in the core tends

to twist quite a bit before reaching 8M, so under parallel transport

the phase of the tensor Sv becomes almost random Thus there is(Illite a bit of cancellation in the visual average, so the strainSV(p) isHlrlall. This establishes inflexibility for a deformation, and the resultfor mappings follows in dilllcnsion three using the Beltrami equation.The proof of geometric illtlexibility will rely on properties of thecouformal strain /:"'11, t,h(~ ViHlIH,] exteuHion ex('lJ) and the visual dis-

26 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

tortion M V developed in Appendices A and B We will assume miliarity with this material throughout this section

fa-Definitions Let M =]H[nIr be a hyperbolic manifold, n ~ 3 A

deformation of r (or of M) is a vector field v on 860-1 such that

T'*(v) - v is a conformal vector field for alIT' E r. A deformation is

trivial if v is conformal Two deformations VI and V2 are equivalent

ifVI - V2 is conformal

A deformation can be thought of as an infinitesimal map gatingr to another Kleinian group The trivial deformations corre-spond to moving r by conjugacy inside Isom(IHr)

conju-A deformation is quasiconformal if V is a quasiconformal vectorfield; that is, ifV is continuous and its conformal strain8v is in L oo

as a distribution (On the Riemann sphere this condition is the same

as 118vll 00 < 00.) By considering the eigenspaces of the strain tensor

Sv, we see a nontrivial quasiconformal deformation determines a invariant k-plane field on the sphere for some k, 0< k <n-1 Thus

f-we have the infinitesimal form of Theorem 2.10:

Proposition 2.12 Let M = IHrIr be a hyperbolic manifold whose injectivity radius is bounded above Then any quasiconformal defor- mation of M is trivial.

To explore the effect of a quasiconformal deformation v on the geometry of M =]H[nIf, let V =ex(v) be the visual extension of v

to IHr (§B.1) ThenT*(V) - V is an infinitesimal isometry of IHr foreachT' Er. The extended vector field V is volume-preserving, so itsconformal strain 8V also measures its distortion of the hyperbolicmetric Since the strain of an isometric vector field is zero, the tensor

SV is r-invariant, and therefore it descends to a strain.field on M

which we continue to denote by8V.

To illustrate the idea of inflexibility, we first show:

Proposition 2.13 Suppose p lies in the convex core K of a bolic n-manifold M whose injectivity radius is bounded above by Rl

hyper-on K and below byRo > 0 at p Let v be a quasiconformal tion of M, and let V = ex(v) Then

deforma-118V(p)II ~ 6(d(p, oK)) · 118v(p)1100

where 6(r) + 0 as r -+ 00, and 6(r) depends only on (n, Ro, R

:l.4 GEOMETRIC INFLEXIBILITY 27

Proof If not, we can find a sequence of hyperbolic manifolds Mi =

IHI''''/ri with points Pi in their convex cores Ki' and deformations Vi

Hnch that IISVi(p)II = 1, IISvilloo is bounded by a constant k, and

tl(pi,8K i ) +- 00. (Here lti = ex(vi).) Lift to the universal cover sothatPi = 0 is the origin in the ball model for hyperbolic space; then11·<;Vi(O) II = 1

Since d(O,8Ki) +- 0, the limit set of ri converges to the wholeHphere in the Hausdorff topology The injectivity radius of Mi is

hounded above on Ki and below at Pi, so by Proposition 2.4 we

cnn pass to a subsequence such that ri converges geometrically to a

I(leinian groupr whose limit set is the whole sphere The injectivityradius of M = IHrn/r is bounded above by Rl' so M admits no(1'lasiconformal deformations

Now by Corollary A.I!, the space of k-quasiconformal vectorfields, modulo conformal vector fields, is compact in the topology oftlniform convergence on the sphere Thus after correcting by con-f()rmal vector fields (to obtain equivalent deformations), we may

",Iso assume Vi converges uniformly to a k-quasiconformal vectorfi(~ld v. Then V is a deformation of r, and v is nontrivial because1I."VV(O)1f = limIISVi(O)11 = 1, where V = ex(v). This contradictsrigidity ofr and establishes the Proposition

•

The preceding Proposition bounds the strain deep in the coreill terms of the strain on the sphere at infinity The main step int.he proof of inflexibility is the next Lemma, showing a bound ont.he distortion over a large finite sphere gives an improved bound att.lle center of the sphere When iterated, this improvement yields(~xponentialdecay of deformations; and when integrated, the boundsf()r deformations give bounds for mappings

The visual distortion M v :IHrn +-IRis defined by

Mv(p) = inf IIv - wlloo(p),

sw=o

where Ilv - wlloo(p) denotes the maximum length of the vector field

(HI v modulo confornlal vector fields Thus Mv is r-invariant and it

t.oo deHcends to a funct.ioll Oil fl.:f.

28 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

Although we are mostly interested in bounding IISV(p)ll, we will

do so by first bounding the visual distortion Since tt = Sv

deter-mines v up to a conformal vector field, M v really only depends onJ.L

- but in a rather implicit way The use of the visual distortion M v

to measure the size ofJ.L is crucial; the proof breaks down with manyother natural measurements of the size ofJ.L.

Let S(p,r) denote the hyperbolic sphere of radius r centered at

p.

Lemma 2.14 (Geometric decay) Let v be a deformation of a perbolic manifold M = JHrl/r with convex core K Suppose the in- jectivity radius of M is bounded above by Rl on K and below by

hy-Ro >0 at p Then there is a radius r(n, Ro, R 1 ) such that whenever

uni-a sequence of Kleiniuni-an groups ri, a sequence of deformationsVi and

a sequence of radiiri -. 00,such that MVi(P) >1/2 but MVi(q) $ 1

on the sphere S(p, ri) C Ki. The bounds on the injectivity radiusimply that after passing to a subsequence, ri tends geometrically

to a Kleinian group r whose limit set is the whole sphere, and theinjectivity radius ofM = JHrl/r is bounded above byR 1 •

After passing to a subsequence and correcting by conformal vectorfields, we can assume Vi converges uniformly to a quasiconformalvector fieldv (see Corollary B.18) Since the convergence is uniform,

Mv(P) = limMvi(P) ~ 1/2 But since M is rigid, v is conformal,

and therefore Mv =O The Lemma follows· by contradiction

•Next we show the visual extension of a deformation tends to an

(infinitesi~al)isometry exponentially fast in the convex core

Remark on notation Here and in the sequel, en and C~ denote

constants that depend only on the dimension n Different occurrences

of these constants are meant to be independent

2.4 GEOMETRIC INFLEXIBILITY 29

Theorem 2.15 (Infinitesimal inflexibility) Let M be a bolic n-manifold, n 2:: 3 Suppose the injectivity radius of M in its convex core K ranges in the interval [Ro,R 1], whereRo > o.

hyper-Let V =ex(v) be the visual extension of a quasiconformal '1nation v of M Then for any p E M we have:

defor-IISV(p)II ~ CnMv(p) ~ C~exp(-ad(p,M- K)) IISvlloo

Ilere a >0 depends on (n,Ro,Rl).

Proof By Theorem B.I5, we have

'rhus we need only establish the second inequality in the statement

ofthe Theorem, and we may assume p E K

Let r be the radius guaranteed by Lemma2.14 for the constants

(Ro,R 1) Let N be the largest integer such that d(p, f:) K) ~ N r

rrhen we can apply Lemma 2.14 N times to conclude that

1

Mv(P) ~ 2N sup Mv(q).

QES(p,Nr)

Now 1/2 N ~ 2exp(-ad(p,8K)) wherea = (log2)/r, and Mv(q) ~

(]nIlSvlloo by TheoremB.15, so the stated bound on Mv(p) follows

II

To givea global version of the preceding result, we need to show

a point deep in the convex core remains reasonably deep after a(Illasi-isometry

Proposition 2.16 Let ~ :JHrl ~ JHrl be an L-quasi-isometry, and

let A be a closed subset ofS~-l. Then q,(hull(A)) is contained within

a d(n, L)-neighborhood ofhull(q,(A))

Proof Suppose the origin in tIle ball model for hyperbolic space

iH contained in the convex hull of A Then there exist two points

~I;,X'E AwhORe an~llln.r Hf'pnrntion is at least tr/4; otherwise A (and

30 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

its convex hull) would be contained in a hemisphere Thus any point

PE hull(A) lies within a universally bounded distancedo of a geodesic, c hull(A) Now <p(,) is a quasi-geodesic with endpoints in <I>(A),

so it lies within distance d 1 (n, L) of a geodesic " c hull(<I> (A) ) (see,e.g [Thl, Prop 5.9.2]), and therefore

d(p, M - K) < d(p, x) ~ Ld(4)(p) , <p(x))

< L(d(q>(p) , M' - K') +d(n, L)).

Solving for d(<p(p) , M' - K') gives the Corollary

The argument whenep(p) ¢ K' is similar

•

Proof of Theorem 2.11 (Geometric inflexibility) Let M =

IHI3Ir, let M' = lH[3Ir', and let ~ :lH[3 -+ lH[3 denote a lift of'If tothe universal cover Then the boundary values of 'If give a K(L)-

quasiconformal map 'l/; : S~ -+ S~ conjugating r to r'. ApplyingTheorem B.22, we can construct a Beltrami isotopy <Pt such that

4;0 = id, 4;1 = 'l/;, and cPt conjugates r to a Kleinian group ft - TheBeltrami isotopy is the integral of a quasiconformal vector field Vt

satisfying 118vtll ~ k(L).

2.4 GEOMETRIC INFLEXIBILITY 31

Now apply the visual extension to obtain a time-dependent vector

field ltt = ex(Vt) on]HI3. The integral of this vector field gives a family

ofvolume-preserving quasi-isometries ~t : lHI3 -+ IHI3 prolonging ¢t

(by Theorem B.21) The quasi-isometry constant of tI>t is boundedl)y a constant L' depending only L This isotopy of lHI3 descends to

a family of maps M ~ M t =JH[3/rt which we will also denote by<Pt.Since 4>1 = 'l/J, tI>1 :M + M 1 = M' is homotopic to w. To com-plete the proof, we will bound the quasi-isometry constant L(tI>1,p).

Let K t denote the convex core of M t By Theorem B.21, thequasi-isometry constant is bounded by the integral of the strain of

By the preceding Corollary, there is a constant d( L') such that

32 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

2.5 Deep points and differentiability

A quasiconformal conjugacy between a pair of Kleinian groups isoften nowhere differentiable on the limit set

In this section we will show a conjugacy is sometimes forced to

be differentiable and conformal at many points This conformalitycan be thought of as a remnant of Mostow rigidity when the limitset is not the whole sphere It says the fine structure in the limit set

is unchanged by a quasi-isometric deformation

Definitions Let AC S~-l be a compact set, and let K C un be itsconvex hulL We say x E A is a deep pointofAif there is a geodesicray

I :[0,00) -+ K, parameterized by arclength and terminating at x, such that for some

6> 0,

d(l'(s),oK) 2::6s >0

for all s. In other words, the depth of I inside the convex hull

of A increases linearly with hyperbolic length When quantitativeprecision is required we say x is a6-deep point

In terms of the sphere at infinity, a point x E Ais deep if andonly if the blowups ofAabout x converge exponentially fast to the

sphere in the Hausdorff metric on compact sets Equivalently, let

B(x,r) be the spherical ball of radiusr about x, and let s(r) denotethe radius of the largest ball contained in B(x,r) - A Then x is

deep if and only if there is a f3 > 0 such that s(r) ~ r 1 + f3 for all r

for all tEe sufficiently small We may now state:

Theorem 2.18 (Deep conformality) Let M = JH[3/r be a bolic 3-martijold whose injectivity radius is bounded above and below

hyper-in its convex core, and let¢ be a quasiconformal conjugacy from r to another Kleinian groupr'. Then ¢ is'C 1 + a -conjormal at every deep point in the limit '1etA

2.5 DEEP POINTS AND DIFFERENTIABILITY 33

More precisely, if the injectivity radius in the core ranges in

I'lo, R 1 ], cP is K -quasiconformal and x is a 6-deep point, then ¢ is

(,11+0: conformal at x, whereQ >0 depends only on (Ro, Rl' K, 8).

(:>roof. The proof follows the same lines as the proof of geometricinflexibility (Theorem 2.11)

We will work in the upper half-space model ]ffi3 =C x lR+ withcoordinates (z, t). Let ,,(s) = (0,e- S

) denote the geodesic ray

start-illg at (0, 1) at terminating at z = o. Let K be the convex hull ofthe limit set ofr.

By a conformal change of coordinates, we can arrange that the

(leep limit point x is at the origin z = 0, that 1(0) E K, and that

d(,,(s) , 8K) 2:: 6s > 0for all 8 > O By conjugatingr', we can also arrange that cP fixes 0,

I and 00.

Next we embed¢in a Beltrami isotopy¢t,fixing 0,1 and00,with

</)0 = id and ¢1 = ¢, using Theorem B.22 The isotopy cPt integrates

Itcontinuous vector field Vt with 118Vtlloo ~ k, where k depends only

on the dilatation of4> Let vt = eX(Vt), and integrate \It to obtain aquasi-isometric isotopy tPt ofJH[3 prolonging ¢t. Each mapping~t isa.n L-quasi-isometry, where Ldepends only on k.

The mapping4>tU<Pt onS~UlHI3conjugatesr to a Kleinian group

I't o Let K t denote the convex hull of the limit set At ofrt

We claim that

d(/(s),8K t ) 2::6's - d

for all s > 0, where d,6' > 0 are independent of t. Indeed, since(I>t is an L-quasi-isometry, Rt = ~t("[O,00)) is a uniformly quasi-

~eodesic ray, starting near,,(0) and terminating at z =O Thus R t

is contained in a uniformly bounded neighborhood of Ro, and thepoint tPt('Y(s')) closest to ,,(s) satisfies s' > s/L - 0(1). Applying

(~orollary2.17 to estimate the change in the convex hull gives

34 CHAPTER 2 RIGIDITY OF HYPERBOLIC MANIFOLDS

SincefJ?t is a quasi-isometry, the given upper and lower bounds onthe injectivity radius ofM in its convex core provide similar bounds

for M t = IHI 3/ft Combining Theorem 2.15 with the estimate on thedistance to the convex hull boundary just obtained, we conclude thatthe visual distortion ofVt tends to zero exponentially fast along thegeodesic ray 'Y That is,

for some C,Q >0 independent oft.

By the normalization of the Beltrami isotopy, we have Vt(O) = Vt(l) = Vt(oo) = O Thus the exponentially decay of the visualdistortion implies, by Theorem B.26, that Vt is 01+0: at the origin.More precisely,

IVt(z) - v~(O)zl ~C'lz11+0:

whenIzi :::; 1, whereC' is independent oft. Applying Theorem B.27,

we conclude that cP1 isC1+a:-conformal at the origin, as claimed

is a quasiconformal map that is conformal on an open set n We

will show that conformality persists at points x E an that are surrounded by fl This well-surroundedness is guaranteed when x is

well-a deep point of nwell-and8n is shallow.

Some applications to Kleinian groups and iterated rational mapsare given in examples

Definitions A closed set AC S~-l is R-shallow if its convex hull

in lHrn cont.ains no ball of radius R We say Ais shallow if it is

R-shallow for some R > O The terminology is suggested by the factthat a shallow set has no deep points

It is easy to see the following are equivalent: