Uniform twisted homological stability

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.