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30
A rigidity theorem for the solvable BaumslagSolitar groups
, 1996
"... this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory. ..."
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Cited by 51 (10 self)
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this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory.
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
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 49 (8 self)
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2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25
A hyperbolic–by–hyperbolic hyperbolic group
"... Abstract. Given a short exact sequence of finitely generated groups 1 → K → G → H → 1 it is known that if K and G are word hyperbolic, and if K is nonelementary, then H is word hyperbolic. In the original examples due to Thurston, as well as later examples due to Bestvina and Feighn, the group H is ..."
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Cited by 22 (4 self)
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Abstract. Given a short exact sequence of finitely generated groups 1 → K → G → H → 1 it is known that if K and G are word hyperbolic, and if K is nonelementary, then H is word hyperbolic. In the original examples due to Thurston, as well as later examples due to Bestvina and Feighn, the group H is elementary. We give a method for constructing examples where all three groups are nonelementary. Given a short exact sequence of finitely generated groups 1 → K → G → H → 1 it is shown in [Mos96] that if K and G are word hyperbolic and K is nonelementary (that is, not virtually cyclic), then H is word hyperbolic. Since the appearance of that result, several people have asked whether there is an example where all three groups are nonelementary. In all previously known examples the group H is elementary. For instance, let K = π1(S) whereSis a closed hyperbolic surface, let f: S → S be a pseudoAnosov homeomorphism, and let Mf be the mapping torus of f, obtained from S × [0, 1] by identifying
Bounded geometry for Kleinian groups
 Invent. Math
"... Abstract. We show that a Kleinian surface group, or hyperbolic 3manifold with a cusppreserving homotopyequivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a po ..."
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Cited by 21 (3 self)
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Abstract. We show that a Kleinian surface group, or hyperbolic 3manifold with a cusppreserving homotopyequivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a oncepunctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Contents
Finite Subdivision Rules
 Conform. Geom. Dyn
, 2001
"... . We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision rule and a planar tiling associated to it, we obtain an infinite sequence of tilings by recursively sub ..."
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Cited by 19 (7 self)
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. We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision rule and a planar tiling associated to it, we obtain an infinite sequence of tilings by recursively subdividing the given tiling. We wish to determine when this sequence of tilings is conformal in the sense of Cannon's combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be repaced by a single axiom which is implied by either of them, and that it su#ces to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed. This paper is concerned with recursive subdivisions of planar complexes. As an introductory example, we present a finite subdivision rule in Figure 1. There are two kinds of edges and three kinds of tiles. A thin edge is subdivided into five su...
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
Negatively Curved Groups Have The Convergence Property I
 ANNALES ACADEMIAE SCIENTIARUM FENNICAE
, 1995
"... It is known that the Cayley graph \Gamma of a negatively curved (Gromovhyperbolic) group G has a welldefined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 ..."
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Cited by 16 (2 self)
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It is known that the Cayley graph \Gamma of a negatively curved (Gromovhyperbolic) group G has a welldefined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 , then G is topologically conjugate to a cocompact Fuchsian group.
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
Combinatorial and geometrical aspects of hyperbolic 3manifolds, Kleinian groups and hyperbolic 3manifolds
 Soc. Lecture Note Ser
"... This is the edited and revised form of handwritten notes that were distributed with the lectures that I gave at the workshop on Kleinian Groups and Hyperbolic 3Manifolds in Warwick on September 1115 of 2001. 1 The goal of the lectures was to expose some recent work [Min] on the structure ..."
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Cited by 11 (1 self)
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This is the edited and revised form of handwritten notes that were distributed with the lectures that I gave at the workshop on Kleinian Groups and Hyperbolic 3Manifolds in Warwick on September 1115 of 2001. 1 The goal of the lectures was to expose some recent work [Min] on the structure