On the Howe--Moore property for automorphism groups of buildings

Abstract

Let \(G<Aut(X)\) be a totally disconnected locally compact group acting strongly transitively on a locally finite building \(X\) of finite-rank and minimal non-spherical type. For sufficiently large thickness, every weakly mixing strongly continuous unitary representation of \(G\) is \(C_0\). Consequently, if \(G\) has no non-trivial finite-dimensional unitary representations, then \(G\) has the Howe--Moore property. More concretely, this applies to rank-three compact-hyperbolic crystallographic types of thickness \(q+1\) for \(q\geq 19379\), if there are no compact quotients. As an application, we prove that the corresponding Caprace--Rémy Kac--Moody lattices in these types, which are known to be finitely presented simple and Kazhdan, are character-rigid: their extremal characters are only the regular and the trivial character. Consequently they also have no non-trivial invariant random subgroups.