Low-dimensional linear representations of mapping class groups

Abstract

Recently, John Franks and Michael Handel proved that, for $g\ge 3$ and $n\le 2g-4$, 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 $n\le 2g-1$, 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 $n\le 2g-1$.

A section on Aut$F_n$ and two corollaries are added