Geometry & Topology

Asymptotic geometry of the mapping class group and Teichmüller space

Jason A Behrstock

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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 tree-graded in the sense of Druţu and Sapir; this tree-grading has several consequences including answering a question of Druţu 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 δ–hyperbolic. Although for higher complexity surfaces these spaces are not δ–hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmüller space for arbitrary surfaces.

Article information

Geom. Topol., Volume 10, Number 3 (2006), 1523-1578.

Received: 15 August 2005
Revised: 15 July 2006
Accepted: 26 September 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 30F60: Teichmüller theory [See also 32G15] 57M07: Topological methods in group theory 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

mapping class group Teichmüller space curve complex asymptotic cone


Behrstock, Jason A. Asymptotic geometry of the mapping class group and Teichmüller space. Geom. Topol. 10 (2006), no. 3, 1523--1578. doi:10.2140/gt.2006.10.1523.

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