Asymptotic geometry of the mapping class group and Teichmueller space

Abstract

In this work, we study the asymptotic geometry of the mapping class group and Teichmueller 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 tree-graded in the sense of Drutu and Sapir; this tree-grading 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 Teichmueller space, with the Weil-Petersson metric, is delta-hyperbolic. Although for higher complexity surfaces these spaces are not delta-hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmueller space for arbitrary surfaces.

This is the version published by Geometry & Topology on 21 October 2006

References in corpus (10)

• Geometry of the Complex of Curves I: Hyperbolicity
• Bounded cohomology of subgroups of mapping class groups
• Asymptotic geometry of the mapping class group and Teichmueller space
• Geometry of the Weil-Petersson completion of Teichmüller space
• Pants Decompositions of Surfaces
• The automorphism group of the pants complex
• Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
• Tree-graded spaces and asymptotic cones of groups
• Quasiflats with holes in reductive groups
• Surface bundles versus Heegaard splittings

Cited by in corpus (79)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Asymptotic geometry of the mapping class group and Teichmueller space
• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Contracting Boundaries of CAT(0) Spaces
• Convex cocompactness and stability in mapping class groups
• Hierarchically hyperbolic spaces II: Combination theorems and the distance formula
• Boundaries and automorphisms of hierarchically hyperbolic spaces
• On metric relative hyperbolicity
• Strongly Contracting Geodesics in Outer Space
• Median structures on asymptotic cones and homomorphisms into mapping class groups
• Curve complexes are rigid
• Constructing group actions on quasi-trees and applications to mapping class groups
• Thickness, relative hyperbolicity, and randomness in Coxeter groups
• Subfactor projections
• Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity
• A Metrizable Topology on the Contracting Boundary of a Group
• Hierarchical hyperbolicity of graphs of multicurves
• Divergence of CAT(0) Cube Complexes and Coxeter Groups
• A note on subfactor projections
• On train track splitting sequences
• A simple criterion for non-relative hyperbolicity and one-endedness of groups
• The augmented marking complex of a surface
• Geodesics in the mapping class group
• Lacunary hyperbolic groups
• The normal closure of big Dehn twists, and plate spinning with rotating families
• Stable cubulations, bicombings, and barycenters
• Quasi-isometry rigidity of groups
• Dehn filling Dehn twists
• Right-angled Artin groups and Out(F_n) I: quasi-isometric embeddings
• Embedding mapping class groups into finite products of trees
• Divergence and quasimorphisms of right-angled Artin groups
• Recurrent Weil-Petersson geodesic rays with non-uniquely ergodic ending laminations
• Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
• Stable commutator length on mapping class groups
• A characterization of higher rank symmetric spaces via bounded cohomology
• Right-angled Artin groups as normal subgroups of mapping class groups
• Dimension and rank for mapping class groups
• Relatively hyperbolic groups: geometry and quasi-isometric invariance
• Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups
• A survey on Morse boundaries & stability
• Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms
• The geometry of the disk complex
• Convergence and divergence of Kleinian surface groups
• Non-existence of boundary maps for some hierarchically hyperbolic spaces
• Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups
• Uniform models and short curves for random 3-manifolds
• Morse elements in Garside groups are strongly contracting
• Morse geodesics in torsion groups
• The asymptotic geometry of the Teichmüller metric: Dimension and rank
• Divergence in lattices in semisimple Lie groups and graphs of groups
• Tight trees and model geometries of surface bundles over graphs
• Explicit equivalences between CAT(0) hyperbolic type geodesics
• Hyperbolicity via Geodesic Stability
• Groups acting on tree-graded spaces and splittings of relatively hyperbolic group
• Some group theory problems
• Divergence of geodesics in Teichmuller space and the mapping class group
• A recipe for short-word pseudo-Anosovs
• Sublinearly Morse Boundary II: Proper geodesic spaces
• Separating Pants Decompositions in the Pants Complex
• Counting pseudo-Anosovs as weakly contracting isometries
• Markov chains on hyperbolic-like groups and quasi-isometries
• A Cartan-Hadamard type result for relatively hyperbolic groups
• Random walks and contracting elements I: Deviation inequality and Limit laws
• The Asymptotic Cone of Teichmüller Space: Thickness and Divergence
• Divergence in right-angled Coxeter groups
• Hyperbolic quasi-geodesics in CAT(0) spaces
• Extensions of multicurve stabilizers are hierarchically hyperbolic
• Uniform uniform exponential growth of subgroups of the mapping class group
• Pseudo-Anosov mapping classes from pure mapping classes
• Multitwists in big mapping class groups
• Divergence, thickness and hypergraph index for general Coxeter groups
• Asymptotic cones of snowflake groups and the strong shortcut property
• The coarse geometry of hexagon decomposition graphs
• Universal tree-graded spaces and asymptotic cones
• Asymptotic dimensions of the arc graphs and disk graphs
• The model theory of the curve graph
• The geometry of subgroup embeddings and asymptotic cones
• Mapping Class Groups and Interpolating Complexes: Rank
• Characterizations of Stability via Morse Limit Sets