Homological stability for general linear groups over Dedekind domains

TL;DR

This paper establishes a new homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, extending previous stability results to include all stabilisation maps.

Contribution

It introduces a generalized stability theorem that accounts for all stabilisation maps, not just those by free modules of rank 1, and applies to reductive Borel--Serre spaces.

Findings

Homological stability holds for automorphism groups over Dedekind domains.

Stability applies to all stabilisation maps, not only rank-1.

Results extend to Clausen and Jansen's reductive Borel--Serre spaces.

Abstract

We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only stabilisation by the free module of rank 1. We show the same kind of stability holds for Clausen and Jansen's reductive Borel--Serre spaces.