Separating subgroups of mapping class groups in homological representations

Abstract

Let $\Gamma$ be either the mapping class group of a closed surface of genus $\geq 2$, or the automorphism group of a free group of rank $\geq 3$. 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.