Modular representations via the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups

TL;DR

This paper investigates modular representations derived from the first homology of certain subgroups of mapping class groups and automorphism groups of free groups, revealing their composition factors and stability properties.

Contribution

It introduces a detailed analysis of the modular representations from homology of congruence subgroups and establishes periodic stability results, advancing understanding of these algebraic structures.

Findings

Computed composition factors and multiplicities of the representations

Established periodic representation stability results

Analyzed modules of coinvariants for the Torelli group's abelianization

Abstract

We study sequences of modular representations of the symplectic and special linear groups over finite fields obtained from the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups, and the module of coinvariants for the abelianization of the Torelli group. In all cases we compute the composition factors and multiplicities for these representations, and obtain periodic representation stability results in the sense of Church--Farb.