Arithmetic representations of mapping class groups

TL;DR

This paper investigates the conditions under which the image of a natural homomorphism from the centralizer of a finite group action on a surface to the symplectic group has finite index or exhibits the Putman-Wieland property, linking geometric and algebraic structures.

Contribution

It provides new criteria for when the image of the centralizer homomorphism is of finite index and when it satisfies the Putman-Wieland property, advancing understanding of mapping class group representations.

Findings

Established a sufficient condition for the image to be a finite index subgroup.

Identified a weaker condition ensuring the Putman-Wieland property holds.

Connected geometric automorphisms with algebraic symplectic representations.

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).