Unitary $L^{p+}$-representations of almost automorphism groups
Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze-Stein property, and let $π$ be a unitary representation of $H$. We show that the representation $\widetildeπ$ of $G$ induced from $π$ is an $L^{p+}$-representation if and only if $π$ 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.
math.RT↗