A unified proof of the howe-moore property

We provide a unified proof of all known examples of locally compact groups that enjoy the Howe{Moore property, namely, the vanishing at infinity of all matrix coefficients of the group unitary representations that are without non-zero invariant vectors. These examples are: connected, non-compact, simple real Lie groups with finite center, isotropic simple algebraic groups over nonarchimedean local fields and closed, topologically simple subgroups of Aut(T) that act 2-transitively on the boundary θT , where T is a bi-regular tree with valence ≥ 3 at every vertex.