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322
Shrinkwrapping and the taming of hyperbolic 3manifolds
 J. Amer. Math. Soc
"... Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite. ..."
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Cited by 83 (0 self)
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Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite.
Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems
, 2004
"... ..."
The type of the classifying space for a family of subgroups
 J. Pure Appl. Algebra
"... We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact su ..."
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Cited by 55 (28 self)
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We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra, for the FarrellJones Conjecture about the algebraic Kand Ltheory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.
Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces
"... A metric space X has Markov type 2, if for any reversible finitestate Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is d ..."
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Cited by 41 (24 self)
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A metric space X has Markov type 2, if for any reversible finitestate Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its biLipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type 2 (in particular, Lp for p> 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp. 1
K.Whittlesey, Bestvina’s normal form complex and the homology of Garside groups, Geom. Dedicata 105
, 2004
"... Abstract. A Garside group is a group admitting a finite lattice generating setD. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the latticeD ..."
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Cited by 39 (2 self)
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Abstract. A Garside group is a group admitting a finite lattice generating setD. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the latticeD, and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π,1)s enjoy Bestvina’s weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually abelian. 1.
Asymptotic geometry of the mapping class group and Teichmüller space
 GEOMETRY & TOPOLOGY
, 2006
"... In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. ..."
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Cited by 33 (6 self)
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In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is treegraded in the sense of Dru¸tu and Sapir; this treegrading has several consequences including answering a question of Drutu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmüller space, with the Weil–Petersson metric, is ı–hyperbolic. Although for higher complexity surfaces these spaces are not ı–hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmüller space for arbitrary surfaces.
Billiards on RationalAngled Triangles
 Comment. Math. Helv
, 1998
"... this paper. The lattice polygons are those polygons for which the surface M ` has a large affine automorphism group (see section 3). In this paper we introduce techniques to analyze affine automorphism groups and apply these techniques to the surfaces, M ` , associated to acute and right rational tr ..."
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Cited by 33 (1 self)
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this paper. The lattice polygons are those polygons for which the surface M ` has a large affine automorphism group (see section 3). In this paper we introduce techniques to analyze affine automorphism groups and apply these techniques to the surfaces, M ` , associated to acute and right rational triangles. In particular we find all acute lattice triangles whose angles are rational with denominator less than 10,000.
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
Thick metric spaces, relative hyperbolicity, and quasiisometric rigidity
, 2005
"... Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasiisometric image of a nonrelatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group bei ..."
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Cited by 30 (9 self)
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Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasiisometric image of a nonrelatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasiisometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasiisometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples