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65
Embeddings Of Gromov Hyperbolic Spaces
 Geom. Funct. Anal
"... . It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasiisometric 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 m ..."
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Cited by 52 (5 self)
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. It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasiisometric 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 quasiisometries. 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...
Genericcase complexity and decision problems in group theory, preprint
, 2003
"... Abstract. We give a precise definition of “genericcase 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 lineartime genericcase complexity. We prove such theorems by ..."
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Cited by 48 (22 self)
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Abstract. We give a precise definition of “genericcase 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 lineartime genericcase complexity. We prove such theorems by using the theory of random walks on regular graphs. Contents 1. Motivation
H.: Poincar'e inequalities and quasiconformal structure on the boundary of some hyperbolic buildings
 Proc. Amer. Math. Soc
"... Abstract. In this paper we shall show that the boundary ∂Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincaré type inequalities. Then by using HeinonenKoskela’s work, we shall prove Loewner capacity estimates for some families of curves of ∂Ip,q and the fact that every quas ..."
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Cited by 33 (1 self)
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Abstract. In this paper we shall show that the boundary ∂Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincaré type inequalities. Then by using HeinonenKoskela’s work, we shall prove Loewner capacity estimates for some families of curves of ∂Ip,q and the fact that every quasiconformal homeomorphism f: ∂Ip,q − → ∂Ip,q is quasisymmetric. Therefore by these results, the answer to questions 19 and 20 of Heinonen and Semmes (Thirtythree YES or NO questions about mappings, measures and metrics, Conform Geom. Dyn. 1 (1997), 1–12) is NO. 1.
The classification of puncturedtorus groups
 ANNALS OF MATH
, 1999
"... 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 puncturedtorus grou ..."
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Cited by 30 (3 self)
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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 puncturedtorus groups. These are free twogenerator 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 puncturedtorus groups (including Bers ’ conjecture that the quasiFuchsian groups are dense in this space) and prove a rigidity theorem: two puncturedtorus groups are quasiconformally conjugate if and only if they are topologically conjugate.
Laminations in Holomorphic Dynamics
, 1998
"... 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 3orbifold quotient of a Kleinian group. To build this object we replace the notion of manifold ..."
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Cited by 27 (3 self)
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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 3orbifold 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) 1794.
Recognizing constant curvature discrete groups in dimension 3
 Trans. Amer. Math. Soc. CMP 97:15
"... Abstract. We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space H 3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negat ..."
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Cited by 22 (9 self)
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Abstract. We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space H 3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the SullivanTukia theorem on groups which act uniformly quasiconformally on the 2sphere. 1.
The Isomorphism Problem for Toral Relatively Hyperbolic Groups
"... We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."
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Cited by 21 (7 self)
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We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree 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 nmanifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsionfree relatively hyperbolic group with abelian parabolics is
Genericity, the ArzhantsevaOl’shanskii method and the isomorphism problem for onerelator groups
"... Abstract. We apply the method of ArzhantsevaOl’shanskii to prove that for an exponentially generic (in the sense of Ol’shanskii) class of onerelator 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 ..."
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Cited by 20 (12 self)
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Abstract. We apply the method of ArzhantsevaOl’shanskii to prove that for an exponentially generic (in the sense of Ol’shanskii) class of onerelator 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 mgenerator nrelator groups where every group has only one Nielsen equivalence class of mtuples generating nonfree subgroups. We also prove that all groups in this class are coHopfian. 1.
Accidental parabolics and relatively hyperbolic groups, preprint
"... 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 re ..."
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Cited by 17 (4 self)
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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 3manifold. 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 grouptheoretic 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 wordhyperbolic 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