Geometry of the mapping class groups III: Quasi-isometric rigidity

TL;DR

This paper proves the quasi-isometric rigidity of the mapping class group of a surface and provides a new proof for the homological dimension of its asymptotic cone, deepening understanding of its geometric structure.

Contribution

It establishes the quasi-isometric rigidity of mapping class groups and offers a novel proof for the homological dimension of their asymptotic cones.

Findings

Mapping class groups are quasi-isometrically rigid.

Homological dimension of the asymptotic cone is 3g-3+m.

Provides a new proof for Behrstock and Minsky's result.

Abstract

Let S be an oriented surface of finite type of genus g with m punctures and where 3g-3+m>1. We show that the mapping class group M(S) of S is quasi-isometrically rigid. We also give a different proof of the following result of Behrstock and Minsky: The homological dimension of the asmyptotic cone of M(S) of S equals 3g-3+m.