The universal group of Burger--Mozes and the Howe--Moore property

TL;DR

This paper investigates the Howe--Moore property of the universal group $U(F)^+$ associated with Burger--Mozes, revealing it does not have this property when $F$ is primitive but not 2-transitive, and establishing new characterizations and properties.

Contribution

It introduces a new unitary representation to show $U(F)^+$ lacks the Howe--Moore property under certain conditions and characterizes when it does have this property, also proving the relative Howe--Moore property.

Findings

$U(F)^+$ lacks Howe--Moore property when $F$ is primitive but not 2-transitive.

Provides a characterization of $U(F)^+$ having the Howe--Moore property.

Establishes $U(F)^+$ has the relative Howe--Moore property.

Abstract

By constructing a new unitary representation we prove the universal group $U(F)^+$ of Burger--Mozes does not have the Howe--Moore property when $F$ is primitive but not $2$-transitive. It is well known $U(F)^+$ does have this property when $F$ is $2$-transitive. Along the way, we give a characterization of the universal group, when $F$ is primitive, to have the Howe--Moore property, and also prove $U(F)^+$ has the relative Howe--Moore property. 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.