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

Abstract

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

Ver. 2, added Remark 5.1 on the relation of Theorem 2 with buildings; 15 pages, comments are welcome!