Bounded cohomology of groups acting on trees with almost prescribed local actions

TL;DR

This paper investigates the bounded cohomology of groups acting on trees with specific local actions, showing vanishing results under certain conditions and infinite dimensionality in others, revealing complex cohomological behavior.

Contribution

It establishes new vanishing and non-vanishing results for bounded cohomology of groups acting on trees with prescribed local actions, depending on transitivity properties.

Findings

Vanishing of bounded cohomology when $F'$ is 2-transitive.

Infinite dimensional second bounded cohomology when $F'$ is not 2-transitive.

Differentiates cohomological behavior based on local action transitivity.

Abstract

We prove the vanishing of bounded cohomology of the groups acting on trees with almost prescribed local actions $G(F, F')$, where $F<F'$ are finite permutation groups such that $F'$ is 2-transitive. By contrast, when $F'$ is not 2-transitive, we prove that the second bounded cohomology with real coefficients of the groups $G(F, F')$ is infinite dimensional.