A note on bounded-cohomological dimension of discrete groups

Abstract

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite boundedcohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.

Bounded cohomology and binate groups

“…In the proof of Theorem 1.3, it will be useful to extend the definition of UBC from groups to group homomorphisms [39,Definition 4.5].…”

Transgression in bounded cohomology and a conjecture of Monod

“…On the other hand, a classical result due to Johnson [34] asserts that the bounded cohomology of an amenable group vanishes in all positive degrees. Moreover, Löh [38] recently found non-amenable groups whose bounded cohomology with trivial real coefficients vanishes in all positive degrees, and Bucher and Monod [9] proved a similar statement for SL 2 over non-Archimedian local fields. These latter results have in common that the bounded cohomological dimension of the respective group is zero.…”

Lamplighters and the bounded cohomology of Thompson's group

“…Many classical "large" transformation groups satisfy these conditions, starting with the group of compactly supported homeomorphisms of R n which was proved to be boundedly acyclic by Matsumoto-Morita [MM85]; the latter result was recently widely generalised in [Löh17] and [FFLM21a].…”

“…First, we should recall that many examples of boundedly acyclic groups (with trivial coefficients) have been discovered, starting with the theorem of Matsumoto-Morita [MM85]. Recent examples include, among others, [Löh17], [FFLM21a], [FFLM21b], [MN21]. A very nice general criterion, but for degree two only, is given in [FFL21].…”