Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification

TL;DR

This paper investigates the action of the mapping class group on homology of surface covers, constructing a faithful infinite-dimensional representation that captures the Nielsen-Thurston classification, with applications to braid groups and free group automorphisms.

Contribution

It introduces a homological representation of the mapping class group that detects Nielsen-Thurston types and extends the theory to braid groups and free groups.

Findings

Constructed a faithful infinite-dimensional homological representation.

Showed the representation detects Nielsen-Thurston classification.

Discussed applications to braid groups and automorphisms of free groups.

Abstract

We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.