Quantitative representation stability over linear groups

TL;DR

This paper develops a new technique for proving quantitative representation stability for sequences of linear group representations, with applications to homology of congruence subgroups and stability of mapping class groups.

Contribution

It introduces a novel method for establishing representation stability and applies it to key problems in algebraic topology and group theory, partially answering open questions.

Findings

Proves a vanishing result for higher syzygies of VIC- and SI-modules.

Establishes new homological stability results for mapping class groups.

Provides partial resolutions to questions by Church and Putman-Sam.

Abstract

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of a regularity theorem of Church-Ellenberg in the context of FI-modules. We apply these techniques to the rational homology of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups. This partially resolves a question raised by Church and Putman-Sam. We also prove new homological stability results for mapping class groups and automorphism groups of free groups with twisted coefficients.