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327
Orbihedra Of Nonpositive Curvature
 Progress in Mathematics
, 1995
"... . A 2dimensional orbihedron of nonpositive curvature is a pair (X; \Gamma), where X is a 2dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and \Gamma is a group of isometries of X which acts properly disc ..."
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Cited by 155 (7 self)
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. A 2dimensional orbihedron of nonpositive curvature is a pair (X; \Gamma), where X is a 2dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and \Gamma is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X; \Gamma) which turns out to depend only on \Gamma and prove that, if X is boundaryless, then either (X; \Gamma) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. 1. Introduction The idea of considering curvature bounds on metric spaces belongs to Alexandrov [Ale], Busemann [Bus] and Wald [Wal]. Busemann initiated the theory of spaces of nonpositive curvature. Later, Bruhat and Tits [BrTi] showed that there is a natural ...
Strong rigidity of II1 factors arising from malleable actions of weakly rigid groups, I
"... Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1 ..."
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Cited by 75 (15 self)
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Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1
Topology of homology manifolds
 Ann. of Math
, 1996
"... The study of the localglobal geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space th ..."
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Cited by 53 (14 self)
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The study of the localglobal geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space theory opened new perspectives. He constructed important examples of 3dimensional homology manifolds with nonmanifold points, which led to the study of other structural properties of these spaces, and also established his shrinking criterion that can be used to determine when homology manifolds obtained as decomposition spaces of manifolds are manifolds [4]. In the 1970s, the fundamental work of Cannon and Edwards on the double suspension problem led Cannon to propose a conjecture on the nature of manifolds, and generated a program that culminated with the EdwardsQuinn characterization of higherdimensional topological manifolds as ENR homology manifolds satisfying a weak general position property known as the disjoint disks property [17, 26,23]. Starting with the work of Quinn [45, 47], a new viewpoint has emerged. Recent advances [10] use techniques of controlled topology to produce a wealth of previously
The Novikov conjecture and groups with finite asymptotic dimension
, 1995
"... this paper we shall prove the coarse BaumConnes conjecture for proper metric spaces with nite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures. Theorem 1.1 Let be a ..."
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Cited by 52 (5 self)
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this paper we shall prove the coarse BaumConnes conjecture for proper metric spaces with nite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures. Theorem 1.1 Let be a nitely presented group whose classifying space B
Geometry of the complex of curves. II. Hierarchical structure
 MW02] [Nag88] [O’N83] [Rie05] [Sar90] [Thu88] [Tro92] Howard Masur and Michael
"... 2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25 ..."
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Cited by 50 (8 self)
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2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25
The classification of Kleinian surface groups I: models and bounds
, 2002
"... Abstract. We give the first part of a proof of Thurston’s Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a “Lipschitz model ” for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topologi ..."
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Cited by 49 (3 self)
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Abstract. We give the first part of a proof of Thurston’s Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a “Lipschitz model ” for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick part, and to give apriori geometric bounds. Contents
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
Random walks on infinite graphs and groups  a survey on selected topics
 Bull. London Math. Soc
, 1994
"... 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 ..."
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Cited by 34 (2 self)
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2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4
Isoperimetric and isodiametric functions of groups
 Ann. of Math
"... This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C2 2Cm for some constant C> 0 then n α is ..."
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Cited by 33 (15 self)
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This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C2 2Cm for some constant C> 0 then n α is equivalent (“big O”) to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is n 3/4α. On the other hand if n α is equivalent to the Dehn function of a finitely presented group then the first m digits of α are computable in time ≤ C2 22Cm for some constant C. This implies that, say, functions n π+1, n e2 and n α for all rational numbers α ≥ 4 are equivalent to the Dehn functions of some finitely presented group and that n π and n α for all rational numbers α ≥ 3 are equivalent to the smallest isodiametric functions of finitely presented groups.