Unitary $L^{p+}$-representations of almost automorphism groups

Abstract

Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze–Stein property, and let $\pi $ be a unitary representation of $H$. We show that the representation $\widetilde{\pi }$ of $G$ induced from $\pi $ is an $L^{p+}$-representation if and only if $\pi $ is an $L^{p+}$-representation. We deduce the following consequence for a large natural class of almost automorphism groups $G$ of trees: For every $p \in (2,\infty )$, the group $G$ has a unitary $L^{p+}$-representation that is not an $L^{q+}$-representation for any $q < p$. This in particular applies to the Neretin groups.