Simplicial bounded cohomology and stability

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.Comment: 53 pages. Comments welcome