Low dimensional linear representations of the mapping class group of a nonorientable surface

Abstract

Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$, then $f$ factors through the abelianization of $\mathcal{M}(N_{g,n})$, which is ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ for $g\in \{5,6\}$ and ${\mathbb{Z}}_2$ for $g\ge 7$. If $g\ge 7$, $n=0$ and $m=g-1$, then either $f$ has finite image (of order at most two if $g\ne 8$), or it is conjugate to one of four "homological representations". As an application we prove that for $g\ge 5$ and $h<g$, every homomorphism $\mathcal{M}(N_{g,0})\to \mathcal{M}(N_{h,0})$ factors through the abelianization of $\mathcal{M}(N_{g,0})$.

References in corpus (2)

• Homomorphisms between mapping class groups
• Crosscap slides and the level 2 mapping class group of a nonorientable surface

Cited by in corpus (1)

• On finite index subgroups of the mapping class group of a nonorientable surface