Bounded Cohomology of Groups acting on Cantor sets

TL;DR

This paper proves that the full homeomorphism group of the Cantor set and Thompson's group V are boundedly acyclic, meaning their bounded cohomology with trivial coefficients vanishes in positive degrees, highlighting a new universal property for these groups.

Contribution

It establishes the bounded acyclicity of the full homeomorphism group of the Cantor set and Thompson's group V, introducing a potential unifying approach for groups acting on the Cantor set.

Findings

Both groups are boundedly acyclic.

Thompson's group V is the first finitely generated, type F_infinity group with this property.

The results suggest a broader class of groups may share this bounded acyclicity.

Abstract

We study the bounded cohomology of certain groups acting on the Cantor set. More specifically, we consider the full group of homeomorphisms of the Cantor set as well as Thompson's group $V$. We prove that both of these groups are boundedly acyclic, that is the bounded cohomology with trivial real coefficients vanishes in positive degrees. Combining this result with the already established $\mathbb{Z}$-acyclicity of Thompson's group $V$, will make $V$ the first example of a finitely generated group, in fact the first example of a group of type $F_\infty$, which is universally boundedly acyclic. Before proving bounded acyclicity, we gather various properties of the groups under consideration and certain subgroups thereof. As a consequence the proofs of bounded acyclicity will be relatively short. It will turn out that the approaches to handle these groups are very similar. This suggests that there could be a unifying approach which would imply the bounded acyclicity of a larger class of groups acting on the Cantor set, including the discussed ones.