Arithmetic representations of mapping class groups

Abstract

Let $S$ be a closed oriented surface and $G$ a finite group of orientation preserving automorphisms of $S$ whose orbit space has genus at least $2$. There is a natural group homomorphism from the $G$-centralizer in $Diff^+(S)$ to the $G$-centralizer in $Sp(H_1(S))$. We give a sufficient condition for its image to be a subgroup of finite index and a weaker condition for this to have no finite nonzero orbit (the Putman-Wieland property).