Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces

Abstract

Consider a compact locally symmetric space $M$ of rank $r$, with fundamental group $\Gamma$. The von Neumann algebra $\vn(\Gamma)$ is the convolution algebra of functions $f\in\ell_2(\Gamma)$ which act by left convolution on $\ell_2(\Gamma)$. Let $T^r$ be a totally geodesic flat torus of dimension $r$ in $M$ and let $\Gamma_0\cong\bb Z^r$ be the image of the fundamental group of $T^r$ in $\Gamma$. Then $\vn(\Gamma_0)$ is a maximal abelian $\star$-subalgebra of $\vn(\Gamma)$ and its unitary normalizer is as small as possible. If $M$ has constant negative curvature then the Puk\'anszky invariant of $\vn(\Gamma_0)$ is $\infty$.