Uniform twisted homological stability

TL;DR

This paper establishes a homological stability theorem for certain families of discrete groups with stable ranges independent of representation choice, confirming predictions for moments of quadratic L-functions over function fields.

Contribution

It introduces a novel stability result applicable to various groups with coefficients in algebraic representations, advancing the understanding of homological stability in this context.

Findings

Proves homological stability with stable range independent of representation.

Confirms predictions for moments of quadratic L-functions over function fields.

Extends previous work to a broader class of groups and representations.

Abstract

We prove a homological stability theorem for families of discrete groups (e.g. mapping class groups, automorphism groups of free groups, braid groups) with coefficients in a sequence of irreducible algebraic representations of arithmetic groups. The novelty is that the stable range is independent of the choice of representation. Combined with earlier work of Bergstr\"om--Diaconu--Petersen--Westerland this proves the Conrey--Farmer--Keating--Rubinstein--Snaith predictions for all moments of the family of quadratic $L$-functions over function fields, for sufficiently large odd prime powers.