The relative Howe-Moore property for the universal group U(F)^+

The aim of this article is twofold: firstly, we give a characterization of the universal group U(F)^+, with F being primitive, to have the Howe-Moore property. Secondly, we prove that U(F)^+ has the relative Howe-Moore property, when F is primitive. These two results are a consequence of a strengthening of Mautner's phenomenon for locally compact groups acting on d-regular trees and having Tits' independence property.