---

APPEARED IN BULLETIN OF THE

arXiv:math/9210224v1 [math.GT] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 207-216RIEMANN SURFACES AND THE GEOMETRIZATIONOF 3-MANIFOLDSCURT MCMULLEN1. IntroductionAbout a decade ago Thurston proved that a vast collection of 3-manifolds carrymetrics of constant negative curvature.

These manifolds are thus elements of hy-perbolic geometry, as natural as Euclid’s regular polyhedra. For a closed manifold,Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos],so topology and geometry mesh harmoniously in dimension 3.This remarkable theorem applies to all 3-manifolds, which can be built up inan inductive way from 3-balls, i.e., Haken manifolds.

Thurston’s construction ofa hyperbolic structure is also inductive. At the inductive step one must find theright geometry on an open 3-manifold so that its ends may be glued together.

Usingquasiconformal deformations, the gluing problem can be formulated as a fixed-pointproblem for a map of Teichm¨uller space to itself. Thurston proposes to find thefixed point by iterating this map.Here we outline Thurston’s construction and sketch a new proof that the itera-tion converges.

Our argument rests on a result entirely in the theory of Riemannsurfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a mapof lesser dilatation) when lifted to a sufficiently large covering space (e.g., the uni-versal cover). This contraction gives an immediate estimate for the contraction ofThurston’s iteration.A detailed account of these results appears in [Mc1, Mc2]; therefore, we haveadopted a more informal tone here.

An expository sequel relating hyperbolic man-ifolds and iterated rational maps appears in [Mc3].The setup for Thurston’s theorem is discussed at length in [Mor],andThurston’s proof is contained in the series [Th3, Th5, Th6, Th7], etc. See also[Th2] for a survey and an introduction to the more general geometrization conjec-ture and [Th1] for more on hyperbolic geometry and Kleinian groups, as well asthe existence of hyperbolic structures on many manifolds which are not Haken.Background in Teichm¨uller theory can be found, e.g., in [Bers2, Bers4, Gard].2.

Teichm¨uller theory and quasiconformal mapsIn this section we state results from [Mc1] concerning contraction of canonicalmappings between Teichm¨uller spaces.Received by the editors October 28, 1991. Presented to the Symposium on Frontiers of Math-ematics, New York, 14–19 December 1988.1991 Mathematics Subject Classification.Primary 57M99, 30C75.Research partially supported by an NSF Postdoctoral Fellowship.c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2CURT MCMULLENLet f : X →Y be an orientation-preserving homeomorphism between two Rie-mann surfaces X and Y .If f is smooth then we can measure its conformal distortion as follows. First,consider a real-linear isomorphism L : C →C.

Think of the complex structureon C as being recorded by the family of circles centered at the origin, since theseare the orbits of multiplication by the unit complex numbers. The image of thesecircles under L is a family of ellipses of constant eccentricity.

The ratio of majorto minor axes gives a number K(L) ≥1 called the dilatation of L.We define the dilatation K(f) as the supremum of the dilatation of the derivativeDf : Tx →Tf(x), over all tangent spaces Tx to X.The dilatation of f is 1 if and only if f is conformal.If K(f) < ∞, f isquasiconformal.Technical remark. The natural degree of smoothness to require is that the dis-tributional derivatives of f lie in L2.Assuming f is quasiconformal, we can try to adjust it by isotopy to minimize itsdilatation.

By compactness of maps with bounded dilatation, at least one extremal(dilatation minimizing) quasiconformal map exists in each isotopy class. For exam-ple, if X and Y are tori, any extremal map is an affine stretch (it takes the formz 7→Az + Bz + C in the universal covers of X and Y ).Teichm¨uller’s theorem describes the extremal maps when X and Y are hyperbolicRiemann surfaces of finite area (equivalently, surfaces of negative Euler characteris-tic obtained from compact surfaces by possibly removing a finite number of points).In each isotopy class there is a unique extremal Teichm¨uller map.

Away from afinite number of singularities, the map is again an affine stretch in appropriate localcoordinates.Relaxation of quasiconformal maps. Now consider quasiconformal mappingson the unit disk ∆={z: |z| < 1}.Any quasiconformal map f : ∆→∆extends to a homeomorphism of S1, theboundary of the disk.

Of course f is always isotopic to the identity; to obtain aninteresting extremal problem, we require that the isotopy fix the values of f on S1.Our first result states that a Teichm¨uller mapping is no longer extremal whenlifted to the universal cover.Theorem 2.1. Let f : X →Y be a Teichm¨uller mapping between hyperbolic Rie-mann surfaces of finite type.

Then the map ˜f : ∆→∆obtained by lifting f to theuniversal covers of X and Y is not extremal among quasiconformal maps with thesame boundary values (unless f is conformal).This result can be reformulated in terms of natural maps between Teichm¨ullerspaces. Given a Riemann surface X, the Teichm¨uller space Teich(X) consists ofequivalence classes of data (f : X →X1) where X1 is another Riemann surface andf is a quasiconformal map.

Given two points (f : X →X1) and (g : X →X2),the Teichm¨uller distance d(X1, X2) is defined to be inf log(K(h)) where h rangesover all quasiconformal maps h : X1 →X2, which are isotopic to g ◦f −1 (rel idealboundary). Thus d(X1, X2) = 0 exactly when there is a conformal isomorphismin the correct isotopy class, and in this case we consider (f, X1) and (g, X2) torepresent the same point in Teichm¨uller space.

RIEMANN SURFACES AND THE GEOMETRIZATION OF 3-MANIFOLDS3Now given a covering Y →X, there is a natural inclusion Teich(X) ֒→Teich(Y );one simply forms the corresponding covering space of each Riemann surface qua-siconformally equivalent to X. For the special case of the universal covering of aRiemann surface of finite type, Theorem 2.1 says the mapTeich(X) →Teich(∆)is a contraction for the Teichm¨uller metric.Poincar´e series.

To check that a map between Teichm¨uller spaces is a contraction,it suffices to show the derivative of the map is a contracting operator. It is actuallymore convenient to work with the coderivative, dual to the derivative operator.Let Q(X) denote the space of holomorphic quadratic differentials φ(z)dz2 on X,such that||φ|| =ZX|φ(z)| |dz|2 < ∞.With the above norm, Q(X) is a Banach space.Q(X) may be naturally identified with the cotangent space to Teichm¨uller spaceat X; its norm is the infinitesimal form of the Teichm¨uller cometric.

(As Q(X) gen-erally fails to be a Hilbert space, this is a Finsler metric rather than a Riemannianmetric. Technically, Q(X) is the predual to the tangent space.

)If Y →X is a covering space, then there is a natural push-forward operatorΘ : Q(Y ) →Q(X).This operator is the coderivative at X of the inclusion Teich(X) ֒→Teich(Y ).In the case of the universal covering, the operator Θ is identical with the classicalPoincar´e series operatorΘ(φ)=Xγ∈Γγ∗φ,which takes an integrable quadratic differential φ on the disk and converts it intoan automorphic form for the Fuchsian group Γ = π1(X) (and therefore an elementof Q(X)) [Poin].A more precise formulation of Theorem 2.1 isTheorem 2.2 (Kra’s Theta conjecture). ||Θ|| < 1 for classical Poincar´e series.This means the inclusion of Teichm¨uller spaces is a contraction even at theinfinitesimal level.Amenability.

One can characterize those coverings for which contraction is ob-tained in terms of the purely combinatorial notion of amenability.To introduce this notion, first consider the case of a graph (1-complex) G. Forany set V of vertices of G, let ∂V denote the set of vertices at distance 1 from V . (A vertex at distance 1 is connected to V by an edge but does not itself lie in V .

)Then the expansion γ of G is given byγ=inf |∂V ||V |over all finite sets of vertices V .

4CURT MCMULLENIf the expansion is 0, G is amenable; otherwise it is nonamenable, and the bound-ary of any vertex set is comparable in size to the set itself. For example, there isa unique infinite tree with degree d at each vertex; it is amenable when d = 2 (thetree is an infinite line) and nonamenable for d > 2 (in fact the expansion constantis d −2).Let X be a hyperbolic Riemann surface of finite type; then we can choose afinite graph G ⊂X such that π1(G) surjects onto π1(X).

An amenable coveringp : Y →X is one for which p−1(G) is an amenable graph. It is easy to check thatthe definition is independent of the choice of G.Remark.

Alternatively, a covering is amenable if there exists a linear functional (amean)m : L∞(π1(X)/π1(Y )) →R,invariant under the left-action of π1(X), such that inf(f) ≤m(f) ≤sup(f) for allf. A normal covering is amenable if the deck transformations form an amenablegroup; cf.

[Gre, Pier].Theorem 2.3. Let Y →X be a covering of a hyperbolic Riemann surface of finitetype.

Then either(1) The covering is amenable, ||Θ|| = 1, and the induced map Teich(X) →Teich(Y ) is a global isometry for the Teichm¨uller metric, or(2) The covering is nonamenable, ||Θ|| < 1, and Teich(X) →Teich(Y ) iscontracting.The universal covering is easily shown to be nonamenable, so this theorem con-tains the previous two.The proof of this result is technical, but the relation between relaxation andnonamenability is easy to describe.It is related to the idea of a chain letter or pyramid game. To join the gameon a given round you must (a) pay $1 and (b) get two other people to join onthe next round.

Ten rounds later you leave the game, collecting $1,024 from the“descendents” of your two new members. In real life someone eventually loses, buton a nonamenable graph (which is necessarily infinite), it is possible to enrich everyvertex by drawing capital at a steady rate from infinity.Similarly, when a quasiconformal map is lifted to a cover, we can begin relaxingit on some compact set while creating a certain amount of additional stress near theboundary.

Using nonamenability, the stress can be entirely dissipated to infinity.Bibliographical remarks. Theorem 2.1 was checked in many examples by Strebel[Str]; Ohtake proved ||Θ|| = 1 for abelian coverings [Oh].

Other relations betweenamenability and function theory appear in [Grom, Br, LS].Dependence on moduli. Let us focus again on the case of the universal cov-ering ∆→X.

For application to iteration, it is useful to know how much themap Teich(X) →Teich(∆) contracts the Teichm¨uller metric, because a uniformlycontracting iteration has a fixed point.Theorem 2.4. For the universal covering of a Riemann surface X of genus g withn punctures,||Θ|| ≤C(L, g, n) < 1,

RIEMANN SURFACES AND THE GEOMETRIZATION OF 3-MANIFOLDS5where L is the length of the shortest geodesic on X.This theorem is immediate once ||Θ|| is shown to be a continuous function onthe moduli space Mg,n, by compactness of the set of Riemann surfaces withoutshort geodesics [Mum].Moreover, ||Θ|| →1 as L →0; there is no uniform bound on all of modulispace. Intuitively, a Riemann surface with a short geodesic is degenerating towardsan infinite cylinder, whose fundamental group is Z and whose universal cover isamenable.3.

Hyperbolic 3-manifoldsIn this section we describe Thurston’s theorem on hyperbolic 3-manifolds andhow the preceding results give a new approach to the proof.A hyperbolic 3-manifold N is a complete Riemannian 3-manifold with a metric ofconstant curvature −1. We will only consider N with finitely generated fundamentalgroups.Up to isometry, there is a unique simply connected hyperbolic 3-manifold, hy-perbolic space H3, which is topologically a 3-ball.

Hyperbolic geometry tends toconformal geometry at infinity; for example, H3 may be compactified by the Rie-mann sphere bC in such a way that hyperbolic isometries extend to conformal maps.Since the universal cover of N is isometric to H3, we can think of N as H3/Γwhere Γ is a Kleinian group, i.e., a discrete group of hyperbolic isometries. Thereis a maximal open set Ω⊂bC on which Γ acts properly discontinuously, and fromthis we form the Kleinian manifold (H3 ∪Ω)/Γ.Thus any hyperbolic manifold N is provided with a natural Riemann surfaceboundary ∂N = Ω/Γ.There are two important topological properties of any hyperbolic 3-manifold.First, N is irreducible: any 2-sphere in N bounds a ball, or equivalently π2(N) = 0.This is immediate from contractibility of H3, the universal cover of N.The second property is that N is atoroidal; this means any incompressible torusT 2 ⊂N is peripheral (homotopic to an end of N).

Equivalently, a discrete groupof hyperbolic isometries isomorphic to Z ⊕Z is parabolic, i.e., it is conjugate to alattice of translations acting on bC by< z 7→z + 1,z 7→z + τ >for some choice of τ. Geometrically, such a subgroup determines a finite volumeend of N (a cusp), diffeomorphic to T 2 × [0, ∞) but rapidly narrowing.Thurston’s theorem states that in a large category of 3-manifolds (including,e.g., all 3-manifolds with nonempty boundary), these are the only obstructions toexistence of a hyperbolic structure.Theorem 3.1 (Thurston). An atoroidal Haken 3-manifold is hyperbolic.A Haken manifold is one built up inductively from 3-balls by gluing along in-compressible submanifolds of the boundary.A piece of the boundary is compressible if there is a simple curve, which is es-sential in the boundary but bounds a disk in the manifold.

When gluing together3-manifolds, there is danger of creating a reducible manifold by gluing two com-pressible curves together; the two disks the curve bounds join to form an essential

6CURT MCMULLEN2-sphere. A Haken manifold is always irreducible: by assumption, the gluing locusis incompressible, so this danger never arises.Remark.For simplicity we will suppress consideration of the parabolic locusP ⊂∂M; in general one designates a portion of the boundary which is to be realizedas cusps.

For example, a knot complement frequently carries a hyperbolic structure;the boundary of a tubular neighborhood of the knot is a torus corresponding to arank 2 cusp.Sketch of Thurston’s proof of Theorem 3.1. A finite collection of disjointballs obviously carries a hyperbolic structure.

Start gluing them together alongincompressible submanifolds of their boundary. By an orbifold trick (using An-dreev’s theorem, see [Mor]), one need only deal with the case of gluing along entireboundary components.

At the inductive step, one has a hyperbolic realization N ofa 3-manifold M with incompressible boundary and gluing instructions encoded byan orientation-reversing involution τ : ∂M →∂M. The construction is completedby the following result:Theorem 3.2.

M/τ has a hyperbolic structure if and only if the quotient is ato-roidal.This key result is intimately related to iteration on Teichm¨uller space, and itthis relation that allows the results of §2 to be brought to bear.Let Mdenote a topological (or equivalently smooth) 3-manifold.LetGF(M) denote the space of hyperbolic 3-manifolds N, which are homeomorphic toM.Technical remark.N is required to be geometrically finite, and each N isequipped with a choice of homeomorphism to M up to homotopy equivalence relboundary. Then, as in Teichm¨uller theory, N1 and N2 are regarded as the samepoint in GF(M) if there is an isometry N1 →N2 in the appropriate homotopyclass.A remarkable feature of dimension 3 is the following:Theorem 3.3.

As long as M admits at least one hyperbolic realization, there isa 1-1 correspondence between hyperbolic structures on M and conformal structureson ∂M, i.e.,GF(M) ∼= Teich(∂M). (The isomorphism is by N 7→∂N.)Remarks.

(1) If ∂M is empty then the hyperbolic structure on M is unique; this isMostow rigidity. (2) Let M = S × [0, 1] where S is a surface of genus≥2; then M has at least onehyperbolic realization (consider a Fuchsian group).

The theorem states that givenany two Riemann surfaces X, Y ∈Teich(S) there is a unique 3-manifold interpo-lating between them; this is Bers’ simultaneous uniformization theorem [Bers1]. (3) The general theorem stated above was developed by Bers, Maskit, and othersand put into final form by Sullivan [Sul1].To prove Theorem 3.2, we must find in GF(M) the correct geometry so thatthe ends to be glued together have compatible shape.

This can be formulated as afixed-point problem in Teichm¨uller space, as follows.

RIEMANN SURFACES AND THE GEOMETRIZATION OF 3-MANIFOLDS7First, the skinning mapσ : Teich(∂M) →Teich(∂M)is defined by forming, for each N ∈GF(M), the quasi-fuchsian covering spaces foreach component of ∂N and recording the conformal structures on the new endsthat appear. Then, the gluing instructions determine an isometryτ : Teich(∂M) →Teich(∂M),and a fixed point for τ ◦σ corresponds to a hyperbolic manifold N ∈GF(M), whoseends fit together isometrically under the gluing instructions.Properties of the skinning map.

A detailed treatment of the skinning mapwould take us deeper into the realm of Kleinian groups than we wish to venture atthe moment; here we simply report what we would have learned upon our return. (1) The skinning map typically contracts the Teichm¨uller metric (||dσX|| < 1 ateach point X).

Thus if τ ◦σ has a fixed point X, it can be found by iteration:(τ ◦σ)n(Y ) →Xfor every Y ∈Teich(∂M). (An important exception is the case of a 3-manifoldwhich fibers over the circle; this can be described as (M = F × [0, 1])/τ, where thesurface F is a fiber and τ is the monodromy map.

In this case σ is an isometry anda different approach is required; cf. [Th5].

)(2) In general, the skinning map does not contract uniformly. This is sensible,because M/τ might not admit a hyperbolic structure.

A potential obstruction is acylinder S1×[0, 1] in M, whose ends are glued together by τ to form a nonperipheraltorus. Then (as noted above) M/τ cannot be hyperbolic.

(3) A 3-manifold is acylindrical if every cylinder(S1 × [0, 1], S1 × {0, 1}) ⊂(M, ∂M)with its ends resting on essential curves in ∂M can be deformed into the boundary.Such a manifold will remain atoroidal for any gluing instructions, so there is noevident obstruction to the hyperbolicity of M/τ. One expects the iteration (τ ◦σ)to be robust in this case, and our first result (see [Mc2]) isTheorem 3.4.

If M is acylindrical, the skinning map is uniformly contracting:||dσX|| < C(M) < 1.Thus M/τ has a hyperbolic structure for any choice of gluing map τ.Remark. Thurston has proved the stronger assertion that the image of the skinningmap is bounded in the acylindrical case.

(4) We will briefly sketch how the results of the preceding section bear on thestudy of ||dσ||. Suppose M is acylindrical.

Here is how the map σ appears fromthe point of view of Riemann surfaces. Starting with X in Teich(∂M), one formscountably many copies of its universal cover ˜X.

These are then patched togetherin a complicated but canonical pattern (depending on the topology of M) to formσ(X), the image of X under the skinning map.

8CURT MCMULLENNow suppose X1 and X2 are in Teich(∂M), and let φ : X1 →X2 be an extremalquasi-conformal map. We can form a new quasi-conformal map˜φ : σ(X1) →σ(X2)with the same dilatation as φ, as follows: lift φ to a map between correspondingcopies of the universal covers of X1 and X2 and complete by continuity.By Theorem 2.1, ˜φ can be relaxed to a map of lesser dilatation on each copyof the universal cover without changing its boundary values.Thus the relaxedmaps still fit together, and we have shown that σ contracts the Teichm¨uller metric.Moreover, the refined version of this relaxation result — Theorem 2.4 — yieldsTheorem 3.5.

||dσX|| < c(L) < 1 where L is the length of the shortest geodesicon X.This result even holds in the cylindrical case, so long as M is not an interval bun-dle over a surface. The cylindrical case requires a discussion of the nonamenabilityof covers of X other than the universal cover.

(5) The preceding does not yet yield uniform contraction; rather, it reduces theproblem to a study of short geodesics.This is a significant simplification, however, because a short geodesic controls thegeometry of a hyperbolic manifold over a large distance, by the Margulis lemma.When combined with the theory of geometric limits of quadratic differentials (cf.Appendix to [Mc1]), one finds a qualitative picture which again forces contractionunless the short geodesic lies on one end of a compressing cylinder in the 3-manifold.This cannot occur in the acylindrical case, so then the contraction is uniform.In general we find:Theorem 3.6. For any initial guess Y ∈Teich(∂M), eitherYn=(τ ◦σ)n(Y ) →a (unique) fixed point X,or Yn develops short geodesics, bounding cylinders in M linked by τ to form anonperipheral torus in M/τ.Thus the gluing problem is solvable if and only if M/τ is atoroidal.4.

EpilogueTwo other iterations on Teichm¨uller space have been analyzed with a similarparadigm, and indeed one of our motivations was to give a parallel treatment ofthe skinning map. One iteration is the action on Teich(S) of an element of themapping class group of the surface S [Th4, FLP]; the other arises in the constructionof critically finite rational maps [Th8, DH].The paradigm is to play (a) geometric control in the absence of shortgeodesics, against (b) topological information in their presence.For example, a mapping class whose minimum translation distance is achiev-ed is geometric; it either fixes a point in Teichm¨uller space (and can be representedby an isometry), or it fixes a geodesic (this is the pseudo-Anosov case.) Otherwisethe minimum translation distance is not achieved, so there are Riemann surfacestending to infinity in moduli space and translated a bounded distance.

In the endof moduli space the surfaces have short geodesics, which are necessarily permuted;

RIEMANN SURFACES AND THE GEOMETRIZATION OF 3-MANIFOLDS9thus we have the topological conclusion that the mapping class is reducible. Thisapproach appears in [Bers3].Similarly, given a smooth branched cover f : S2 →S2 whose critical pointseventually cycle, one seeks a conformal structure preserved up to isotopy by f(rel the post-critical set); then f can be geometrized as a rational map.Thisdesired conformal structure can again be described as a fixed point for an iterationon Teichm¨uller space.

The iteration contracts uniformly in the absence of shortgeodesics; in their presence, one locates a topological obstruction to geometrization.Finally, in retrospect it is promising to think of the skinning map as an instance ofrenormalization; a similar approach to the Feigenbaum phenomenon (using infinite-dimensional Teichm¨uller spaces) has been proposed by Douady and Hubbard andpursued by Sullivan [Sul2].Addendum October 1991. The connection with renormalization is now under-stood more precisely; construction of the Feigenbaum fixed-point closely resem-bles the construction of hyperbolic structures on 3-manifolds, which fiber over thecircle—the one case omitted from the discussion above.

This analogy and others(which relate the construction of rational maps to that of hyperbolic 3-manifolds)are discussed in more detail in [Mc3]. Abundant progress towards understandingrenormalization of quadratic polynomials appears in [Sul3].A new approach to the Theta conjecture and its variants is discussed in [BD,Mc4].References[BD]D. Barrett and J. Diller, Poincar´e series and holomorphic averaging, Invent.

Math. (toappear).

[Bers1] L. Bers, Simultaneous uniformization, Bull. Amer.

Math. Soc.

66 (1960), 94–97. [Bers2]Uniformization, moduli and Kleinian groups, Bull.

London Math. Soc.

4 (1972),257–300. [Bers3]An extremal problem for quasiconformal maps and a theorem by Thurston, ActaMath.

141 (1978), 73–98. [Bers4]Finite dimensional Teichm¨uller spaces and generalizations, Bull.

Amer. Math.

Soc.5 (1981), 131–172.[Br]R. Brooks, The fundamental group and the spectrum of the Laplacian, Comment.

Math.Helv. 56 (1985), 581–596.[DH]A.

Douady and J. Hubbard, A proof of Thurston’s topological characterization of rationalmaps, Acta Math. (to appear).[FLP]A.

Fathi, F. Laudenbach, and V. Po´enaru, Travaux de Thurston sur les surfaces, Aster-isque, vol. 66-67, Soc.

Math. France, Paris, 1979.

[Gard] F. Gardiner, Teichm¨uller theory and quadratic differentials, Wiley Interscience, 1987.[Gre]F. P. Greenleaf, Invariant means on topological groups, Van Nostrand, 1969.

[Grom] M. Gromov, Structures m´etriques pour les vari´et´es riemanniennes, Textes Math., vol. 1,CEDIC, Paris, 1981.[LS]T.

Lyons and D. Sullivan, Function theory, random paths and covering spaces, J. Differ-ential Geom. 19 (1984), 299–323.[Mc1]C.

McMullen, Amenability, Poincar´e series and quasiconformal maps, Invent. Math.

97(1989), 95–127. [Mc2]Iteration on Teichm¨uller space, Invent.

Math. 99 (1990), 425–454.

[Mc3]Rational maps and Kleinian groups, Proceedings of the International Congress ofMathematicians Kyoto 1990, Springer-Verlag, 1991, pp. 889–900.

[Mc4]Amenable coverings of complex manifolds and holomorphic probability measures,Invent. Math.

(to appear).

10CURT MCMULLEN[Mor]J. Morgan, On Thurston’s uniformization theorem for three dimensional manifolds, TheSmith Conjecture, Academic Press, 1984, pp. 37–125.[Mos]D.

Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math Stud., vol.

78,Princeton Univ. Press, Princeton, NJ, 1972.

[Mum] D. Mumford, A remark on Mahler’s compactness theorem, Proc. Amer.

Math. Soc.

28(1971), 289–294.[Oh]H. Ohtake, Lifts of extremal quasiconformal mappings of arbitrary Riemann surfaces, J.Math.

Kyoto Univ. 22 (1982), 191–200.[Pier]J.

P. Pier, Amenable locally compact groups, Wiley Interscience, New York, 1984. [Poin] H. Poincar´e, M´emoire sur les fonctions Fuchsiennes, Acta Math.

1 (1882/3), 193–294.[Str]K. Strebel, On lifts of extremal quasiconformal mappings, J. d’Analyse Math.

31 (1977),191–203.[Sul1]D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolicmotions, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony BrookConference, Ann.

of Math. Stud., vol.

97, Princeton Univ. Press, Princeton, NJ, 1981.

[Sul2]Quasiconformal homeomorphisms in dynamics, topology and geometry, Proceed-ings of the International Conference of Mathematicians, 1986, vol. 2 Amer.

Math. Soc.,Providence, RI, 1986, pp.

1216–1228. [Sul3]Bounds, quadratic differentials and renormalization conjectures, Mathematics intothe Twenty-first Century, Amer.

Math. Soc.

Centennial Publications, vol. 2, Amer.

Math.,Soc., Providence, RI, 1992, pp. 417–466.[Th1]W.

P. Thurston, Geometry and topology of three-manifolds, Princeton lecture notes, 1979. [Th2]Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull.Amer.

Math. Soc.

6 (1982), 357–381. [Th3]Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds,Ann.

of Math. (2) 124 (1986), 203–246.

[Th4]On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer.

Math.Soc. 19 (1988), 417–432.

[Th5]Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds whichfiber over the circle, Ann. of Math.

(2) (to appear). [Th6]Hyperbolic structures on 3-manifolds III: Deformation of 3-manifolds with incom-pressible boundary, Ann.

of Math. (2) (to appear).

[Th7]Hyperbolic structures on 3-manifolds IV: Construction of hyperbolic manifolds, inpreparation. [Th8]On the combinatorics and dynamics of iterated rational maps, preprint.Department of Mathematics, Princeton University, Princeton, New Jersey 08544Current address: Department of Mathematics, University of California, Berkeley, California94720E-mail address: ctm@math.berkeley.edu

---

출처: arXiv:9210.224 • 원문 보기