Bounded cohomology and binate groups

Abstract

A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of R n (Matsumoto-Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic.We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups F , T , and V is as simple as possible.

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

“…While our results recover some known computations of second bounded cohomology (see e.g. [30,38,6,26]), the previous proofs of these facts rely on a careful analysis of second cohomology, and in doing so they often appeal to deep theorems which are specific to their setting. The advantage of our approach is that it bypasses the cohomological techniques and only focuses on the bounded part.…”

“…Therefore, a homological analogue of the Witte-Morris Theorem cannot hold. This also provides some evidence that a question of Navas [54,Question 8] Moving to actions on the circle, applying recent results on boundedly acyclic resolutions [26] we are able to deduce the following result from Theorem 1.3: Theorem 1.7. Let G be a group of orientation preserving piecewise linear (or piecewise projective) homeomorphisms of the circle.…”

Finitely generated simple left orderable groups with vanishing second bounded cohomology

“…However all of these examples are indicable and we do not know whether they can be left orderable. Moreover, as mentioned above there exist continuum many countable left-orderable groups which are boundedly acyclic and neither indicable nor locally indicable [8]. We believe that further research into Question 1.4 will lead to a better understanding of the bounded cohomology of finitely generated groups and of left orderable groups.…”

“…It is unknown whether bounded acyclicity is stable under directed unions [8,Section 4.4]. However, in degree 2 it is:…”

“…The first is the hypothesis of finite generation. Indeed, using the construction of Mather [18] it is easy to show that every countable left orderable group embeds in a countable left orderable perfect group which has vanishing bounded cohomology in every positive degree [8]. But such groups are not finitely generatable.…”