A new approach to twisted homological stability, with applications to congruence subgroups

TL;DR

The paper introduces a novel method for proving twisted homological stability, applicable to symmetric and general linear groups, improving stable ranges and extending results to twisted coefficients for various rings.

Contribution

It presents a new, adaptable approach to twisted homological stability, generalizing Borel's theorem to twisted coefficients for many rings.

Findings

Improved stable ranges for symmetric and general linear groups.

Extended Borel's theorem to twisted coefficients.

Method is adaptable to nonstandard situations.

Abstract

We introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the stable range given by the traditional method (due to Dwyer), it is easier to adapt to nonstandard situations. As an illustration of this, we generalize to $GL_n$ of many rings $R$ a theorem of Borel which says that passing from $GL_n$ of a number ring to a finite-index subgroup does not change the rational cohomology. Charney proved this generalization for trivial coefficients, and we extend it to twisted coefficients.