Abstract. We give a precise definition of “generic-case complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory- the word, conjugacy and membership problems- all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs. Contents 1. Motivation

. It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any ffi -hyperbolic metric space embeds isometrically into a complete geodesic ffi -hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. 1. Introduction The study of Gromov hyperbolic spaces has been largely motivated and dominated by questions about Gromov hyperbolic groups. This paper studies the geometry of Gromov hyperbolic spaces without reference to any group or group action. One of our main theorems is 1.1. Embedding Theorem. Let X be a Gromov hyperbolic geodesic metric spa...

. In this paper we shall show that the boundary @I p;q of the hyperbolic building I p;q considered in [1] admits Poincar'e type inequalities. Then by using Heinonen-Koskela's work [7], we shall prove Loewner capacity estimates for some families of curves of @I p;q and the fact that every quasiconformal homeomorphism f : @I p;q \Gamma! @I p;q is quasisymetric. Therefore by these results, the answers to questions 19 and 20 of Heinonen and Semmes [8] are NO. 1. Introduction In recent work Heinonen and Koskela [7] showed that in metric spaces in which the modulus of the family of curves joining two continua is controlled, quasiconformal homeomorphisms are quasisymetric. They characterized such spaces (called Loewner spaces) by the existence of Poincar'e type inequalities. For instance, R n (n 2), Carnot groups, and so the boundary of any non compact symmetric space of rank 1 (and dimension at least 3) are Loewner spaces. In this paper we shall show that the boundary of some hyperbolic...

this paper we explore a construction which attempts to provide an element of the dictionary that has so far been missing: an explicit object that plays for a rational map the role played by the hyperbolic 3-orbifold quotient of a Kleinian group. To build this object we replace the notion of manifold by \lamination", Date: June 25, 1998. Appeared in J. Dierential Geom. 47 (1997) 17-94.

Thurston’s ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers ’ conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate. Contents 1. The ending lamination conjecture and its consequences

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We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is

Abstract. By constructing, in the relative case, objects analoguous to Rips and Sela’s canonical representatives, we prove that the set of conjugacy classes of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite. An important result of W.Thurston is: Theorem 0.1 ([T] 8.8.6) Let S be any surface of finite area, and N any geometrically finite hyperbolic 3-manifold. There are only finitely many conjugacy classes of subgroups G ⊂ π1(N) isomorphic to π1(S) by an isomorphism which preserves parabolicity (in both directions). It is attractive to try to formulate a group-theoretic analogue of this statement: the problem is to find conditions such that the set of images of a group G in a group Γ is finite up to conjugacy. If Γ is word-hyperbolic and G finitely presented, this has been the object of works by M.Gromov ([G] Theorem 5.3.C’) and by T.Delzant [Del], who proves the finiteness (up to conjugacy) of the set of images by morphisms not factorizing through an amalgamation or an HNN extension over a finite group. As a matter of fact, if a group G splits as A ∗C B and maps to a group Γ such that the image

Abstract. We apply the method of Arzhantseva-Ol’shanskii to prove that for an exponentially generic (in the sense of Ol’shanskii) class of one-relator groups the isomorphism problem is solvable in at most exponential time. This is obtained as a corollary of the more general result that for any fixed integers m> 1, n> 0 there is an exponentially generic class of m-generator n-relator groups where every group has only one Nielsen equivalence class of m-tuples generating non-free subgroups. We also prove that all groups in this class are co-Hopfian. 1.

In topology, geometry and complex analysis, one attaches a number of interesting mathematical objects to a surface S. The Teichmüller space T (S) is the parameter space of conformal (or hyperbolic) structures on S, up to isomorphism isotopic to the identity. The Mapping Class Group Mod(S) is the group of auto-homeomorphisms