Asymptotic geometry of the mapping class group and Teichmüller space

Abstract

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 $\delta$–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 Teichmüller space for arbitrary surfaces.