Charmenability of twin building lattices

Abstract

Let $Λ= \mathcal G_A (\mathbb F_q)$ be a minimal simply connected split Kac-Moody group of compact-hyperbolic type, and set $Γ=Λ/Z(Λ)$. We prove that $Γ$ is charfinite when $q\geq 11$, or when its Weyl group is $3$-spherical and $q\geq 4$. Here are a few consequences: the only characters of $Γ$ are the trivial and regular characters, every invariant random subgroup is supported on $\{e\}$ and $Γ$ and these are the only uniformly recurrent subgroups, and every nontrivial unitary $Γ$-representation weakly contains the regular representation. Our approach relies on a recent result due to Caprace and Thom, showing that the topological completions $G_+$ and $G_-$ associated to $Γ$ have the Howe-Moore property. We prove a criterion of charmenability for irreducible lattices with trivial amenable radical in products of Howe-Moore groups acting geometrically on proper hyperbolic spaces. The criterion requires the visual boundaries to be Poisson boundaries of suitable random walks and the corresponding lattice actions to be essentially free.