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27
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
Cannon–Thurston Maps for Trees of Hyperbolic Metric Spaces
 J. Differential Geometry
, 1998
"... Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasiisometrically embedded condition. Let v be a vertex of T. Let (Xv, dv) denote the hyperbolic metric space corresponding to v. Then i: Xv → X extends continuously to a map î: ̂Xv → ̂X. This generalizes a Theorem of Can ..."
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Cited by 30 (1 self)
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Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasiisometrically embedded condition. Let v be a vertex of T. Let (Xv, dv) denote the hyperbolic metric space corresponding to v. Then i: Xv → X extends continuously to a map î: ̂Xv → ̂X. This generalizes a Theorem of Cannon and Thurston. The techniques are used to give a new proof of a result of Minsky: Thurston’s ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian groups are also given. 1.
The BaumConnes conjecture for hyperbolic groups
 Invent. Math
"... Abstract. We prove the BaumConnes conjecture for hyperbolic groups and their subgroups. 1. Introduction. The BaumConnes conjecture states that, for a discrete group G, the Khomology groups of the classifying space for proper Gaction is isomorphic to the Kgroups of the reduced group C ∗algebra ..."
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Cited by 22 (2 self)
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Abstract. We prove the BaumConnes conjecture for hyperbolic groups and their subgroups. 1. Introduction. The BaumConnes conjecture states that, for a discrete group G, the Khomology groups of the classifying space for proper Gaction is isomorphic to the Kgroups of the reduced group C ∗algebra of G [3, 2]. A positive answer to the BaumConnes conjecture would provide a complete solution to the problem of computing higher indices of elliptic
Geometry of the complex of curves I: hyperbolicity
 Invent. Math
, 1999
"... 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 gro ..."
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Cited by 16 (4 self)
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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 autohomeomorphisms
Coherence, local quasiconvexity and the perimeter of 2complexes
, 2002
"... A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2complexes which is introduced here. ..."
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Cited by 15 (4 self)
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A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that onerelator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3manifold groups. The main application is to establish the coherence
Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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Cited by 11 (0 self)
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
Finitely presented subgroups of automatic groups and their isoperimetric functions
 J. London Math. Soc
, 1997
"... Abstract. We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgr ..."
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Cited by 10 (0 self)
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Abstract. We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality. 1. Introduction. Despite
On the definition of word hyperbolic groups
 Math. Z
"... Abstract. Formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups. 1. ..."
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Cited by 7 (1 self)
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Abstract. Formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups. 1.
Greedy embeddings, trees, and euclidean vs. lobachevsky geometry
, 2006
"... A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f: V → X such that for every sourcesink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectiv ..."
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Cited by 6 (0 self)
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A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f: V → X such that for every sourcesink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectivity graphs helps to build distributed routing schemes with compact routing tables. In this paper we take a refined look at greedy embeddings, previously addressed in [1, 2], by examining their description complexity as a key parameter in conjunction with their dimensionality. We give arguments showing that the dimensionality lowerbounds for monotone maps do not extend to greedy embeddings. We prove a unified O(log n) lowerbound on the dimension of nostretch greedy embeddings when the host metric is Euclidean or Lobachevsky geometry. The essence of the lower bound entails showing that lowdimensional spaces lack the topological capacity to realize the embeddings of certain graphs with “hard crossroads. ” This technique might be of independent interest. We develop new methods for building concise embeddings of trees (and some other graphs) in 3dimensional Lobachevsky spaces using recursive applications of hyperbolic isometries guided by caterpillarlike decompositions. Our embeddings improve over prior work [1] by achieving O(κ(T) · log n) description complexity, where κ(T) is the caterpillar dimension. We further demonstrate concise O(log n)dimensional greedy embeddings of trees into Euclidean space using techniques inspired by [3], thereby strengthening our belief and intuition that all O(log n) graphs can be embedded with no stretch in ℓ. ∗ PhD candidate. 2
On The Failure Of The CoHopf Property For Subgroups Of WordHyperbolic Groups
 Israel J. Math
, 1999
"... . We provide an example of a finitely generated subgroup H of a torsionfree wordhyperbolic group G such that H is oneended, and H does not split over a cyclic group, and H is isomorphic to one of its proper subgroups. 1. Introduction AgroupH is said to be coHopfian provided that any injectiv ..."
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. We provide an example of a finitely generated subgroup H of a torsionfree wordhyperbolic group G such that H is oneended, and H does not split over a cyclic group, and H is isomorphic to one of its proper subgroups. 1. Introduction AgroupH is said to be coHopfian provided that any injective endomorphism H # H is an automorphism. Many familiar groups are not coHopfian. For instance, any finitely generated infinite abelian group is not coHopfian. Any free product of two nontrivial groups is not coHopfian. Finite groups are obviously coHopfian, but there are more interesting examples: If M is a closed aspherical ndimensional manifold with #(M) #=0then# 1 M is co Hopfian. Indeed, if # : M # M is acoverofM with # 1 M # = # 1 M,then M is aspherical because M is aspherical, and so M and M are homotopy equivalent. But then M must be a finite cover because Hn ( M,Z 2 )=Hn (M,Z 2 ) #=0. And# must be a degree 1 cover because, #( M)=degree(#)#(M ). One can show s...