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
–hyperbolic.
Although for higher complexity surfaces these spaces are not
–hyperbolic,
we establish the presence of previously unknown negative curvature phenomena in
the mapping class group and Teichmüller space for arbitrary surfaces.
Keywords
mapping class group, Teichmüller space, curve complex,
asymptotic cone
