Rigidity properties of mapping class groups and their random quotients

A theorem of Ivanov states that the mapping class group G of a finite-type surface is also the automorphism group of the curve graph, which encodes the intersection patterns of simple closed curves on the surface. In other words, simplicial automorphisms of the curve graph are "rigid": they all come from homeomorphisms of the surface. This fact can then be used to prove other "rigidity" results, such as quasi-isometric rigidity (if a group is quasi-isometric to G, it is commensurable to G) and the fact that every automorphism of G is inner.

 

In this talk, we first introduce the various characters in our story, and review the literature on these results. Then we show that the same type of properties are enjoyed by "random" quotients of mapping class groups, defined by modding out the n-th steps of certain random walks. Some of this talk is based on joint work with Abbott, Berlyne, Ng, and Rasmussen.