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

Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra VN(Γ) is the convolution algebra of functions f∈ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let Tr be a totally geodesic flat torus of dimension r in M and let Γ0≅Zr be the image of the fundamental group of Tr in Γ. Then VN(Γ0) is a maximal abelian ★-subalgebra of VN(Γ) and its unitary normalizer is as small as possible. If M has constant negative curvature then the Pukánszky invariant of VN(Γ0) is ∞.