Bounded cohomology and binate groups

TL;DR

This paper proves that binate groups are boundedly acyclic, unifying previous examples, and explores their bounded cohomology, including new computations for groups acting on the circle and implications for Thompson groups.

Contribution

It establishes that binate groups are universally boundedly acyclic, providing a unifying framework and new examples in bounded cohomology research.

Findings

Binate groups are boundedly acyclic.

New examples of boundedly acyclic groups are constructed.

Bounded cohomology of certain circle-acting groups is computed.

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 $\mathbb{R}^n$ (Matsumoto--Morita) and mitotic groups (L\"oh). 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.