Normed amenability and bounded cohomology over non-Archimedean fields

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued 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 $K$. Moreover, we prove a bounded-cohomological characterization for discrete groups. The algebraic characterization shows that normed $K$-amenability is a very 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 injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either $K$ has positive characteristic or its residue field has characteristic 0, then the comparison map is injective in all degrees. If $K$ is a finite extension of $\mathbb{Q}_p$, 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. A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg--Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer--Vietoris sequence.