Separating subgroups of mapping class groups in homological representations

Abstract

Let $\Gamma$ be either the mapping class group of a closed surface of genus $\ge 2$, or the automorphism group of a free group of rank $\ge 3$. Given any homological representation $\rho$ of $\Gamma$ corresponding to a finite cover, and any term ${\mathcal{I}}_k$ of the Johnson filtration, we show that $\rho ({\mathcal{I}}_k)$ has finite index in $\rho (\mathcal{I})$, the Torelli subgroup of $\Gamma$. Since $[\mathcal{I}:{\mathcal{I}}_k]=\infty$ for $k>1$, this implies for instance that no such representation is faithful.