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15
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.
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 29 (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
Dimension and rank for mapping class groups
"... Dedicated to the memory of Candida Silveira. Abstract. We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An appl ..."
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Cited by 20 (4 self)
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Dedicated to the memory of Candida Silveira. Abstract. We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of BrockFarb’s Rank Conjecture which asserts that MCG has quasiflats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasiflats in Teichmuller space with the WeilPetersson metric. The coarse geometric structure of a finitely generated group can be studied by passage to its asymptotic cone, which is a space obtained by a limiting process from sequences of rescalings of the group. This has played an important role in the quasiisometric rigidity results of [DS], [KL1] [KL2], and others. In this paper we study the asymptotic cone M ω (S) of the mapping
Relatively hyperbolic groups: geometry and quasiisometric invariance
, 2006
"... In this paper it is proved that relative hyperbolicity is an invariant of quasiisometry. As a byproduct we provide simplified definitions of relative hyperbolicity in terms of the geometry of a Cayley graph. In particular we obtain a definition very similar to the one of hyperbolicity, relying on t ..."
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Cited by 13 (1 self)
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In this paper it is proved that relative hyperbolicity is an invariant of quasiisometry. As a byproduct we provide simplified definitions of relative hyperbolicity in terms of the geometry of a Cayley graph. In particular we obtain a definition very similar to the one of hyperbolicity, relying on the existence for every quasigeodesic triangle of a central left coset of peripheral subgroup.
Thin triangles and a multiplicative ergodic theorem for Teichmüller geometry
"... 1.1. Overview. In this paper, we prove a curvaturetype result about Teichmüller space, in the style of synthetic geometry. 1 We show that, in the Teichmüller metric, “thinframed triangles are thin”—that is, under suitable hypotheses, the variation of geodesics obeys a hyperboliclike inequality. T ..."
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Cited by 6 (0 self)
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1.1. Overview. In this paper, we prove a curvaturetype result about Teichmüller space, in the style of synthetic geometry. 1 We show that, in the Teichmüller metric, “thinframed triangles are thin”—that is, under suitable hypotheses, the variation of geodesics obeys a hyperboliclike inequality. This theorem has applications to
Stable Teichmüller quasigeodesics and ending laminations
 Geom. Topol
"... We characterize which cobounded quasigeodesics in the Teichmüller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path γ in T, we show that γ is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canon ..."
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Cited by 5 (0 self)
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We characterize which cobounded quasigeodesics in the Teichmüller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path γ in T, we show that γ is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over γ is a hyperbolic metric space. As an application, for complete hyperbolic 3–manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically infinite ends of N, a key step in Minsky’s proof of Thurston’s ending lamination conjecture for such manifolds.
Curve complexes, surfaces and 3manifolds
, 2006
"... A survey of the role of the complex of curves in recent work on 3manifolds and mapping class groups. ..."
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Cited by 3 (0 self)
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A survey of the role of the complex of curves in recent work on 3manifolds and mapping class groups.
Relative hyperbolicity and Artin groups
, 2002
"... Abstract. We show that an Artin group G with all mij ≥ 7 is relatively hyperbolic in the sense of Farb with respect to the collection of subgroups 〈ai, aj 〉 (where mij < ∞). 1. ..."
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Cited by 2 (0 self)
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Abstract. We show that an Artin group G with all mij ≥ 7 is relatively hyperbolic in the sense of Farb with respect to the collection of subgroups 〈ai, aj 〉 (where mij < ∞). 1.
Understanding WeilPetersson curvature
, 2008
"... A brief history of the investigation of the WeilPetersson curvature and a summary of Teichmüller theory are provided. A report is presented on the program to describe an intrinsic geometry with the WeilPetersson metric and geodesiclength functions. Formulas for the metric, covariant derivative an ..."
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Cited by 1 (0 self)
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A brief history of the investigation of the WeilPetersson curvature and a summary of Teichmüller theory are provided. A report is presented on the program to describe an intrinsic geometry with the WeilPetersson metric and geodesiclength functions. Formulas for the metric, covariant derivative and formulas for the curvature tensor are presented. A discussion of methods is included. Recent and new applications are sketched, including results from the work of LiuSunYau, an examination of the Yamada model metric and a description of Jacobi fields along geodesics to the boundary.