Representation stability, secondary stability, and polynomial functors

TL;DR

This paper establishes a broad framework for representation and secondary homological stability using polynomial functors, extending classical stability results to new group families with twisted coefficients.

Contribution

It introduces a general stability theorem for polynomial coefficient systems, enabling new stability results for various complex groups and improving existing stability ranges.

Findings

Proves homological stability for hyperelliptic mapping class groups with twisted coefficients

Establishes secondary stability for diffeomorphism groups of surfaces

Improves stable range for linear groups of the sphere spectrum

Abstract

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum.