Richard D. Canary

Ends of hyperbolic 3-manifolds

Let N = H3/F be a hyperbolic 3-manifold which is homeomorphic to the interior of a compact 3-manifold. We prove that N is geometrically tame. As a consequence, we prove that Fs limit set L. is either the entire sphere at infinity or has measure zero. We also prove that Ns geodesic flow is ergodic if and only if LF is the entire sphere at infinity. DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY, STANFORD, CALIFORNIA 94305 Current address: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 E-mail address: dick.canary@math.lsa.umich.edu This content downloaded from 40.77.167.14 on Wed, 15 Jun 2016 05:36:06 UTC All use subject to http://about.jstor.org/terms

A covering theorem for hyperbolic 3-manifolds and its applications

IN THIS paper we will prove a theorem which describes how geometrically infinite ends of topologically tame hyperbolic 3-manifolds cover. In particular, we prove that if E is a geometrically infinite end of a topologically tame hyperbolic 3-manifold fi which covers (by a local isometry) another hyperbolic 3-manifold N, then either the covering projection is finite-to-one on some neighborhood of E or N has a finite cover which fibers over the circle. This theorem generalizes a theorem of Thurston [34] proved for geometrically tame hyperbolic 3-manifolds whose (relative) compact cores have incompressible boundary. The key technical result in this paper, called the filling theorem, is that there exists a neighborhood of any geometrically infinite end such that passing through any point is a convenient simplicial hyperbolic surface in the appropriate homotopy class. As a consequence of the filling theorem we see that for any hyperbolic 3-manifold there exists an upper bound to the injectivity radius at any point in the convex core. Our main application of the covering theorem is a characterization of exactly which covers of an infinite volume, topologically tame hyperbolic 3-manifold N = H3/r are geometrically finite. Associated to any geometrically infinite end of N there is a geometrically infinite peripheral subgroup of I-. We will prove that a finitely generated subgroup f of I is geometrically finite if and only if it does not contain a finite index subgroup of a geometrically infinite peripheral subgroup. We remark that using the original results of Bonahon [S] and Thurston [34] it is possible to understand covers of N whose (relative) compact cores have incompressible boundary. We will also discuss briefly the applications of the covering theorem to understanding algebraic and geometric limits of sequences of Kleinian groups. A discussion of these results in the simpler setting when A has no cusps as well as a discussion of their relationship with various conjectures is given in [9].

Cores of hyperbolic 3-manifolds and limits of Kleinian groups

We study the relationship between the algebraic and geometric limits of a sequence of isomorphic Kleinian groups. We prove that, with certain restrictions on the algebraic limit, the algebraic limit is the fundamental group of a compact submanifold of the quotient of the geometric limit. In particular, we show that the algebraic and geometric limits agree, in the absence of parabolics, if the algebraic limit either has nonempty domain of discontinuity or is not isomorphic to a nontrivial free product of (orientable) surface groups and cyclic groups.

The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(…1(M)) denote the space of (conjugacy classes of) discrete faithful representations of …1(M )i nto PSL 2(C). The components of the interior MP(…1(M)) of AH(…1(M)) (as a subset of the appropriate representation variety) are enumerated by the spaceA(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(…1(M)) and hence a conjectural topological enumeration of the components of AH(…1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(…1(M)) has inflnitely many components. In this paper, we begin a study of the global topology of deformation spaces of Kleinian groups. The basic object of study is the space AH(…1(M)) of marked hyperbolic 3-manifolds homotopy equivalent to a flxed compact 3manifold M. The interior MP(…1(M)) of AH(…1(M)) is very well understood due to work of Ahlfors, Bers, Kra, Marden, Maskit, Sullivan and Thurston. In particular, the components ofMP(…1(M)) are enumerated by topological data, namely the set A(M) of marked, compact, oriented, irreducible 3-manifolds homotopy equivalent to M, while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3-manifolds. Thurston’s Ending Lamination Conjecture provides a conjectural classiflcation for elements of AH(…1(M)) by data which are partially topologi

Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ0(N) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of N. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that A(M) = D(M)(2 − D(M)) if M is not handlebody or a thickened torus. We characterize exactly when A(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.

From the Boundary of the Convex Core to the Conformal Boundary

If N is a hyperbolic 3-manifold with finitely generated fundamental group, then the nearest point retraction is a proper homotopy equivalence from the conformal boundary of N to the boundary of the convex core of N. We show that the nearest point retraction is Lipschitz and has a Lipschitz homotopy inverse and that one may bound the Lipschitz constants in terms of the length of the shortest compressible curve on the conformal boundary.

The conformal boundary and the boundary of the convex core

In this note we investigate the relationship between the conformal boundary at infinity of a hyperbolic 3-manifold and the boundary of its convex core. In particular, we prove that the length of a curve in the conformal boundary gives an upper bound on the length of the corresponding curve in the boundary of the convex core. A hyperbolic 3-manifold N is the quotient of hyperbolic 3-space H by a group Γ of isometries. The “boundary at infinity” of H may be identified with the Riemann sphere Ĉ and Γ extends to act on Ĉ as a group of conformal automorphisms. The domain of discontinuity Ω(Γ) of Γ is the largest Γ-invariant open subset of Ĉ which Γ acts on properly discontinuously. If Γ is not abelian, then Ω(Γ) inherits a hyperbolic metric, called the Poincare metric, which Γ acts on as a group of isometries. One may then consider ∂cN = Ω(Γ)/Γ to be the “conformal boundary at infinity” of the hyperbolic 3-manifold N . The conformal boundary is the topological boundary of N = (H ∪ Ω(Γ))/Γ. The first important result concerning the relationship between the geometry of the conformal boundary and the geometry of N is due to Bers [4] who proved that if each component of Ω(Γ) is simply connected and α is a closed curve in its conformal boundary ∂cN , then the geodesic α ∗ in N in the homotopy class of α has length lN(α ∗) at most twice the length l∂cN(α) of α in the conformal boundary. (If α is homotopic to arbitrarily short closed curves in N , then we say lN (α ∗) = 0.) Canary [8] generalized this to the setting of arbitrary Kleinian groups, proving that given any ǫ > 0 there exists K > 0 such that if Ω(Γ) has injectivity radius bounded below by ǫ (at each point), then if α is a closed curve in ∂c(N), then lN(α ∗) ≤ Kl∂cN (α).

Algebraic convergence of Schottky groups

A discrete faithful representation of the free group on g generators F g into Isom + (H 3 ) is said to be a Schottky group if (H 3 ∪ D Γ )Γ is homeomorphic to a handlebody H g (where D Γ is the domain of discontinuity for Γs action on the sphere at infinity for H 3 ). Schottky space S g , the space of all Schottky groups, is parameterized by the quotient of the Teichmuller space F(S g ) of the closed surface of genus g by Mod 0 (H g ) where Mod 0 (H g ) is the group of (isotopy classes of) homeomorphisms of S g which extend to homeomorphisms of H g which are homotopic to the identity. Masur exhibited a domain O(H g ) of discontinuity for Mod 0 (H g )s action on PL(S g ) (the space of projective measured laminations on S g ), so B(H g ) = O(H g )/Mod 0 (H g ) may be appended to S g as a boundary

Local topology in deformation spaces of hyperbolic 3-manifolds

We prove that the deformation space AH.M/ of marked hyperbolic 3‐manifolds homotopy equivalent to a fixed compact 3‐manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3‐manifolds and Bers slices. 30F40; 57M50 The conjectural picture for the topology of the deformation space AH.M/ of all (marked) hyperbolic 3‐manifolds homotopy equivalent to a fixed compact 3‐manifold M has evolved from one of relative simplicity to one far more complicated in recent years. Indeed, the interior of this space has been well-understood since the late 1970’s. Roughly, components of AH.M/ are enumerated by (marked) homeomorphism types of compact 3‐manifolds homotopy equivalent to M , and each component is a manifold parametrized by natural conformal data. In the last decade, however, a string of results has established that the topology of AH.M/ itself is not well-behaved. In particular, AH.M/ fails to be locally connected when M is an untwisted I ‐bundle over a closed surface (see Bromberg [21] and Magid [43]), and a new conjectural picture in which such pathology is prevalent has replaced the old. The present paper clarifies the role that the geometry and topology of 3‐manifolds associated to points in the boundary of AH.M/ plays in the local topology at such points. In particular, we show that the topology of AH.M/ is well-behaved at many points; if M has incompressible boundary, then AH.M/ is locally connected at “generic” points in the boundary. When M is acylindrical or an untwisted I ‐bundle we obtain finer results.

Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL2(C)-character variety X(M) of π1(M). We study the dynamics of the action of Out(π1(M)) on both AH(M) and X(M). The nature of the dynamics reflects the topology of M . The quotientAI(M) = AH(M)/Out(π1(M)) may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to M and its topology reflects the dynamics of the action.