Simplicial bounded cohomology and stability

TL;DR

This paper develops combinatorial techniques to study simplicial bounded cohomology, proving new stability results for linear groups and automorphism groups by recasting classical arguments in the bounded cohomology setting.

Contribution

It introduces a novel combinatorial framework for simplicial bounded cohomology and applies it to establish new stability results for large classes of linear and automorphism groups.

Findings

Proves slope-1/2 stability for bounded cohomology of linear groups over rings with finite Bass stable rank.

Establishes bounded acyclicity results for semi-simplicial sets in homological stability.

Provides a blueprint for deriving bounded cohomological analogues of classical homological stability results.

Abstract

We introduce a set of combinatorial techniques for studying the simplicial bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We apply these methods to prove several new bounded acyclicity results for semi-simplicial sets appearing in the homological stability literature. Our strategy is to recast classical arguments (due to Bestvina, Maazen, van der Kallen, Vogtmann, Charney and, recently, Galatius--Randal-Williams) in the setting of bounded cohomology using uniformly bounded refinements of well-known simplicial tools. Combined with ideas developed by Monod and De la Cruz Mengual--Hartnick, we deduce slope-$1/2$ stability results for the bounded cohomology of two large classes of linear groups: general linear groups over any ring with finite Bass stable rank and certain automorphism groups of quadratic modules over the integers or any field of characteristic zero. We expect that many other results in the literature on homological stability admit bounded cohomological analogues by applying the blueprint provided in this work.