Unitary $L^{p+}$-representations of almost automorphism groups - Dabeler, Antje; Mai Elkiær, Emilie; Gerasimova, Maria; de Laat, Tim (Comptes Rendus. Mathématique)
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 $\widetilde{π }$ 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 $p \in (2,∞ )$, 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.