Homological stability for automorphism groups

TL;DR

This paper develops a unified framework to prove homological stability for various families of groups using a new construction of spaces with group actions, extending classical results to twisted coefficients and new group families.

Contribution

It introduces a systematic method to establish homological stability with twisted coefficients across multiple classical and novel group families using a braided monoidal structure.

Findings

Homological stability holds with polynomial and abelian twisted coefficients.

Classical stability results are extended to new coefficient systems.

The construction applies to previously unconsidered group families.

Abstract

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability. As a byproduct of our work, we construct the braided analogue of the category FI of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.