Representing some II$_1$ factors in $L^2(Λ\backslash G)$

Abstract

Let $G$ be $PGL(n,F)$, $n \geq 3$, $F$ a certain non-archimedean local field; or let $G$ be $PSL(2,\mathbb{R}) \times \cdots \times PSL(2,\mathbb{R})$. Let $Γ$ be a lattice in $G$, and let $( Λ_n )$ be a sequence of lattices in $G$ satisfying the pointwise limit multiplicity property. In this note, we explain how the pointwise limit multiplicity property can be combined with a generalization of a theorem in \cite{ghj} to give representations of the II$_1$ factor $R Γ$ on a subspace of $L^2(Λ_i \backslash G)$ for some $Λ_i$ in $( Λ_n )$. This extends a result in the author's dissertation \cite{ruthphd}.