Maximal abelian subalgebras of the group factor of an ${\tilde{A}}_2$ group

Abstract

An ${\tilde{A}}_2$ group $\Gamma$ acts simply transitively on the vertices of an affine building $△$. We study certain subgroups ${\Gamma }_0\cong {\mathbb{Z}}^2$ which act on certain apartments of $△$. If one of these subgroups acts simply transitively on an apartment, then the corresponding subalgebra of the group von Neumann algebra is maximal abelian and singular. Moreover the Pukánszky invariant contains a type $I_{\infty }$ summand.