  An $\widetilde A_2$ group $\Gamma$ acts simply transitively on the vertices
of an affine building $\triangle$. We study certain subgroups $\Gamma_0 \cong
{\Bbb Z}^2$ which act on certain apartments of $\triangle$. 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 Puk\'anszky invariant contains a type $I_{\infty}$ summand.
