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

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