Strongly Choquet-Deny groups and invariant strictly convex renormings

Abstract: We introduce and study a strengthening of the notion of Choquet-Deny groups, which we call \emph{strongly Choquet-Deny groups}. We show that a countable group $G$ is strongly Choquet-Deny if and only if for every uniformly bounded representation of $G$ on any separable Banach space $X$, there exists an equivalent $G$-invariant strictly convex norm on $X$. We provide a full classification of finitely generated strongly Choquet-Deny groups and show that a finitely generated group is strongly Choquet-Deny if and only if it is virtually abelian. This fully resolves a question of Mikael de la Salle for finitely generated groups. As a further application, we show that for every uniformly bounded representation of a compact extension of an abelian topological group on any separable Banach space, there exists an equivalent invariant strictly convex norm.