HYPERBOLIC 3-MANIFOLDS AND KLEINIAN GROUPS AT THE

HYPERBOLIC 3-MANIFOLDS AND KLEINIAN GROUPS AT THE CROSSROADS KEN’ICHI OHSHIKA Dedicated to Professor Yukio Matsumoto on the occasion of his sixtieth birthday

1. Introduction By virtue of recent work of Perelman on the geometrisation conjecture for 3-manifolds, it has turned out to be the case that to study the topology of 3-manifolds, what remains to be done is to understand hyperbolic 3-manifolds completely. Naturally, the study of hyperbolic 3-manifolds is closely related to that of Kleinian groups. In this talk, we should like to discuss the present situation and the recent progress (y compris the speaker’s own work) in this field and exhibit the prospects. 2. Finite volume vs. infinite volume The biggest rift in studies of hyperbolic 3-manifolds runs between manifolds of finite volume and of infinite volume. The famous rigidity theorem due to Mostow implies that a complete hyperbolic metric of finite volume on a 3-manifold is, if it exists, unique up to isotopy. On the other hand, if a 3-manifold admits a hyperbolic metric of infinite volume, except for some exceptional cases, there is a non-trivial space of moduli of such structures. Therefore, to study hyperbolic 3-manifolds of finite volume, topological information would be sufficient, while it never is for those of infinite volume. For hyperbolic 3-manifolds of finite volume, obviously the volumes are important invariants. Actually, it is known that for a given real number, there are only finitely many (up to isometry) hyperbolic 3manifolds having that number as their volume. Also, Thurston proved, using his theory of hyperbolic Dehn surgery, that the set of volumes of hyperbolic 3-manifolds has the structure of the ordinal number of ω ω . One of the most important remaining problems is to determine this set completely. For that the first step should be to know what is the smallest number contained in this set. This problem has long been studied, and its solution seems to be very near, but has not been obtained yet. 1

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3. Thurston’s problems in Kleinian group theory In his epoch-making work on uniformisation of Haken manifolds, Thurston revolutionised Kleinian group theory including its aim and scope. In his expository paper appeared in Bull AMS, he listed up unsolved problems and conjectures in Kleinian group theory viewed from the topological viewpoint. Here we reassemble the most important ones of them. (1) (Ahlfors) The limit set of every finitely generated Kleinian group would either be the entire sphere or have measure 0. (2) (Marden) For every finitely generated, torsion-free Kleinian group G, the hyperbolic 3-manifold H3 /G would be homeomorphic to the interior of some compact 3-manifold. (G and H3 /G are said to be topologically tame then.) (3) (Bers-Thurston) Every finitely generated Kleinian group would be an algebraic limit of geometrically finite groups. (4) (Ending lamination conjecture) Topologically tame Kleinian groups would be classfied by topological types and end invariants. (5) (Shottky space problem) Develop a convergence theorem in the Schottky space analogous to the double limit theorem. All of these problems are now solved. We shall discuss them one by one, dividing them into some groups.

4. Ahlfors’ conjecture and Marden’s conjecture Ahlfors conjecture says that the limit set ΛG of every finitely generated Kleinian group G would either be the entire sphere or have measure 0. Ahlfors himself proved that this is the case when G is geometrically finite. Here G is said to be geometrically finite when it has a convex deformation retract having finite volume. Maskit, on the other hand, showed that the conjecture is true for purely loxodromic, free Kleinian group. It was Thurston that first considered Ahfors’ conjecture for general geometrically infinite groups. Supposing that a Kleinian group G only consists of loxodromic elements for simplicity, let M = H3 /G be the corresponding hyperbolic 3-manifold. Since Ahfors’ conjecture is obviously true in the case when M is compact, we assume that M is non-compact. By Scott’s core theorem, there is a compact 3-submanifold C, called a compact core, contained in M such that the inclusion is a homotopy equivalence. The ends of M correspond one-to-one to the boundary components of C. An end e of M is said to be geometrically finite when it has

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a neighbourhood which intersects no closed geodesics, otherwise geometrically infinite. The Kleinian group G is geometrically finite if and only if all the ends of M are geometrically finite. Thurston introduced the notion of geometric tameness for geometrically infinite ends facing incompressible components of ∂C. Definition 4.1. Let S be an incompressible component of ∂C and fix some hyperbolic metric on S. A map f : S → M is said to be a pleated surface when the length metric induced by f from M coincides with that of S, and for any point x on S, there is a geodesic containing x at its interior which is mapped totally geodesically by f . The end e facing S is said to be geometrically (infinite) tame if there is a sequence of pleated surfaces homtopic to the inclusion of S tending to e. Thuston proved in the late 70’s that if every end of M is either geometrically finite or geometrically tame, then Ahlfors’ conjecture is true for G. Let us call such Kleinian groups geometrically tame. Thurston showed if an end e is geometrically tame and faces S, there is a neighbourhood of e which is homeomorphic to S × I. This in particular implies that geometrically tame groups are topologically tame. It was Bonahon that proved that the following purely group-theoretic assumption is sufficient to ensure that Kleinian groups are geometrically tame. Theorem 4.2 (Bonahon). Freely indecomposable Kleinian groups are geometrically tame. Canary, combining Bonahon’s work with the construction of branched coverings due to Gromov-Thurston, proved that Ahlfors’ conjecture is true for topologically tame Kleinian groups. This means that Ahlfors’ conjecture follows from Marden’s conjecture. There has been work relating Ahlfors’ and Marden’s conjectures to Bers-Thurston’s by Ohshika, Canary-Minsky, Evans, Brock-Souto in 90’s. Quite recently, Agol and Calegari-Gabai announced that they solved Marden’s conjecture affirmatively. 5. The ending lamination conjecture The ending lamination conjecture has more to do with the classification problem of Kleinian groups than common properties for Kleinian groups such as the other conjectures listed above deal with. This conjecture is closely related to the problem to determine the topological structures of deformation spaces. Here, we formally define the deformation space of a Kleinian group.

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Definition 5.1. Let G be a Kleinian group. The deformation space of G, denoted by AH(G), is defined to be the set of conjugacy classes of faithful discrete representations of G into P SL2 C. We endow AH(G) with the topology induced from the representation space. It is know, by work of Marden, that if G is geometrically finite, its inclusion into P SL2 C regarded as a point in AH(G) has a neighbourhood consisting only of geometrically finite representations. Furthermore, each of these representations is obtained as a quasi-conformal conju2 2 gate of G, i.e., there is a quasi-conformal homeomorphism f : S∞ → S∞ and the representation is obtained as f Gf −1 . We call Kleinian groups obtained as quasi-conformal conjugates of G quasi-conformal deformations of G, and denote the subspace of AH(G) consisting of quasiconformal deformations by QH(G). 2 We denote the complement of the limit set ΛG in S∞ by ΩG . This set 2 ΩG , called the region of discontinuity, is the maximal subset of S∞ on which G acts properly discontinuously. It was shown by Ahlfors that when G is finitely generated, ΩG /G is a Riemann surface of finite type. The work of Ahlfors, Bers, Marden, Sullivan, among others, implies the following two fundamental theorems. Theorem 5.2. Let q : T (ΩG /G) → QH(G) be a map taking a point m ∈ T (ΩG /G) to a quasi-conformal deformation of G obtained from a quasi-conformal homeomorphism f realising a Teichm¨ uller deformation of the original conformal structure on ΩG /G to that representing m. Then q is a possibly ramified covering. Theorem 5.3. Let G and Γ be purely loxodromic geometrically finite Kleinian groups which are isomorphic each other. Suppose furthermore that (H3 ∪ ΩG )/G is homeomorphic to (H3 ∪ ΩΓ )/Γ. Then there is a quasi-conformal homeomorphism realising a quasi-conformal deformation from G to Γ compatible with the given homeomorphism. Thus the conformal structures at infinity which the quotients of the regions of discontinuity give, classify, together with the hemomorphism type, the geometrically finite Kleinian groups. The situation is better than that: we have a parametrisation of QH(G) as above. It remains to consider the case when G is geomerically infinite. For that, we need the notion of ending lamination introduced by Thurston, which we shall discuss below. Let e be a geometrically infinite end of M = H3 /G, supposing that G is purely loxodromic. We first deal with the case when e faces an incompressible boundary component S of a compact core C.

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Since e is geometrically infinite, any neighbourhood of e intersects some closed geodesic. We can show fairly easily that actually any neighbourhood of e contains a closed geodesic. Consider a system of neighbourhoods U1 ⊃ U2 ⊃ . . . which constitute a base of the neighbourhood system around e, and let γi be a closed geodesic contained in Ui . Since e faces S, the closed geodesic γi is homotopic in M to a closed curve ci on S. Bonahon showed the following. Proposition 5.4. In this situation, we can choose γi so that ci is a simple closed curve on S. The support of projective lamination to which [ci ] converges (passing to a subsequence) depends only on e. This support geodesic lamination is defined to be the ending lamination of e. The ending lamination conjecture says that freely indecomposable Kleinain groups would be completely determined up to conjugacy by the homeomorphism types of the corresponding hyperbolic manifolds, the conformal structures at infinity corresponding to geometrically finite ends, and ending laminations corresponding to geometrically infinite ends. We call these pieces of information, conformal structres at infinity and ending laminations, the end invariants. To generalise the notion of ending laminations to ends facing compressible boundary components, we need to introduce the notion of Mausr domain. Definition 5.5. Let S be a boundary component of a compact 3manifold C, which we assume to have genus at least 2. We define C(S) to be the subset of the measured lamination space ML(S) consisting of weighted disjoint unions of boundaries of compressing discs in C. We set M(S) = {λ ∈ ML(S)|i(λ, µ) > 0, ∀µ ∈ C(S)} and call its image in PL(S) under the canonical projection the Masur domain. Otal showed that by restricting laminations to the Masur domain, ends facing compressible components of a compact core can be dealt with by the same way as the incompressible case. Let e be such an end of M and (γi ) be a sequence of closed geodesics tending to e. Assuming M to be topologically tame, by enlarging a compact core if necessary, γi can be assumed to be homotopic in the complement of the compact core to a closed curve ci on the boundary component S facing e. Canary showed that in this situation, we can choose γi so that ci is a simple closed curve and its projective class converges to a projective lamination contained in the Masur domain as i → ∞

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whose support depends only on e. Thus, we can generalise the ending lamination conjecture to freely decomposable Kleinian groups. The ending lamination conjecture has been solved by Minsky in a series of his papers, partially collaborating Masur, Canary and Brock. 6. The Bers-Thurston conjecture The Bers-Thurston conjecture says that every finitely generated Kleinian group would be an algebraic limit of geometrically finite groups. Provided that the ending lamination is true, this conjecture is reduced to a problem of realising a given end invariant on the boundary of the quasi-conformal deformation space of a geometrically finite group. Such a problem was solved by the speaker in 1990 for freely indecomposable Kleinian groups. For freely decomposable Kleinian groups, a convergence theorem, which was a key ingredient for the proof, had been missing. In 2002, Kleineidam-Souto gave such a convergence theorem. This made it possible for the speaker to complete the proof of the Bers-Thurston conjecture in general. 7. The Schottky space problem An analogue of the double limit theorem for Schottky groups is formulated by Otal as follows. Conjecture 7.1. Let G be a Schottky group. Let q : T (ΩG /G) → QH(G) be the ramified covering which we obtain by the Alfors-Bers theory. Let (mi ) be a sequence in T (ΩG /G) converging in the Thurston compactification to a projective lamination [λ] in the Masur domain. (We regard here ΩG /G as the boundary of the handlebody (H3 ∪ ΩG )/G.) Then (q(mi )) would be relatively compact in AH(G). This conjecture can be generalised to geometrically finite function groups. In this generalised form, we only need to impose the condition as above on the asymptotic behaviour of conformal structures on the invariant component of the region of discontinuity. Kleineidam-Souto solved the conjecture in generalised form under the assumption that λ is arational. The conjecture in general has been solved by Kim-Lecuire-Ohshika. Actually, we have proved a stronger version as follows. Definition 7.2. We set D(S) = {λ ∈ ML(S)|∃η > 0, i(λ, ∂E) > 0for all discs and annuli E}

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Theorem 7.3. Conjecture 7.1 is true even if we weaken the assumption that [λ] lies in the Masur domain to the one that λ ∈ D(S). The same is valid for general (geometrically finite) function groups. References [Ag] I. Agol, Tameness of hyperbolic 3-manifolds, preprint [Ber] L. Bers, On boundaries of Teichm¨ uller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. [Bo] F. Bonahon, Bouts des vari´et´es hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), 71-158. [Ca] R. Canary, Ends of hyperbolic 3-manifolds. J. Amer. Math. Soc. 6 (1993), no. 1, 1–35 [CM] R. Canary and Y. Minsky, On limits of tame hyperbolic 3-manifolds. J. Differential Geom. 43 (1996), no. 1, 1–41. [CG] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds. preprint [KLO] I. Kim, C. Lecuire and K. Ohshika, Convergence of freely decomposable Kleinian groups, preprint [Mar] A. Marden, The geometry of finitely generated kleinian groups. Ann. of Math. (2) 99 (1974), 383–462. [MM] H. Masur and Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1999), no. 1, 103–149. [MiI] Y. Minsky, Bounded geometry for Kleinian groups. Invent. Math. 146 (2001), no. 1, 143–192. [KlS] G. Kleineidam, J. Souto, Algebraic convergence of function groups, Comment. Math. Helv. 77 (2002), 244-269. [OhL] K. Ohshika, Ending laminations and boundaries for deformation spaces of Kleinian groups, J. London Math. Soc. 42 (1990), 111–121. [OhM] K. Ohshika, Kleinian groups which are limits of geometrically finite groups, to appear in Mem. AMS. [OhP] K. Ohshika, Realising end invariants by limits of convex cocompact groups, preprint [Ot1] J.-P. Otal, Courants g´eod´esiques et produits libres, Th`ese d’Etat, Universit´e Paris-Sud, Orsay (1988). [ThLiv] W. Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. [ThB] W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan