Normed amenability and bounded cohomology over non-Archimedean fields

Abstract

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedian field K. To capture the features of classical amenability that induce the vanishing of real bounded cohomology, we introduce the notion of normed K-amenability, of which we prove an algebraic characterization. It implies that normed K-amenable groups are locally elliptic, and it relates an invariant, the norm of a K-amenable group, to the order of its discrete finite p-subquotients, where p is the characteristic of the residue field of K. We also prove a bounded cohomological characterization for discrete groups. The algebraic characterization shows that normed K-amenability is a restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K coefficients. We explore this intuition by studying the comparison map, for which surprisingly general statements are available. We show that if K has positive characteristic or a residue field of characteristic 0, then the comparison map is injective in all degrees. If K is a finite extension of Qp, we classify quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies to finitely presented groups in degree 2. It applies in all degrees to fundamental groups of aspherical CW-complexes with a finite number of cells in each dimension. This can also be justified by looking at bounded cohomology of topological spaces over K, which is naturally isomorphic to cellular cohomology for CW-complexes with the above property. This follows from the fact that bounded cohomology over K is a cohomology theory, except for a weaker version of additivity which is however equivalent for finite unions.