Low-dimensional linear representations of mapping class groups

TL;DR

This paper extends known results on the triviality of low-dimensional linear representations of mapping class groups to larger dimensions and nonorientable surfaces, with applications to automorphism groups of free groups.

Contribution

It generalizes previous theorems to cover cases where the dimension is up to 2g-1, including nonorientable surfaces and certain automorphism groups.

Findings

Homomorphisms from mapping class groups to GL(n,C) are trivial for n ≤ 2g-1.

Results apply to nonorientable surfaces and automorphism groups of free groups.

Extended the range of known triviality results for low-dimensional representations.

Abstract

Recently, John Franks and Michael Handel proved that, for $g\geq 3$ and $n\leq 2g-4$, every homomorphism from the mapping class group of an orientable surface of genus $g$ to $\GL (n,\C)$ is trivial. We extend this result to $n\leq 2g-1$, also covering the case $g=2$. As an application, we prove the corresponding result for nonorientable surfaces. Another application is on the triviality of homomorphisms from the mapping class group of a closed surface of genus $g$ to $\Aut (F_n)$ or to $\Out (F_n)$ for $n\leq 2g-1$.