Separating subgroups of mapping class groups in homological representations

TL;DR

The paper demonstrates that homological representations of mapping class groups and automorphism groups of free groups cannot be faithful on the Johnson filtration subgroups, revealing limitations of these representations in capturing the groups' structure.

Contribution

It proves that homological representations associated with finite covers do not distinguish the Johnson filtration subgroups, showing these subgroups have finite index images and are not faithfully represented.

Findings

Homological representations of the Johnson filtration subgroups have finite index images.

No such representation is faithful on the Johnson filtration subgroups.

The Johnson filtration subgroups have infinite index in the Torelli subgroup.

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.