# Journal of Operator Theory

Volume 36, Issue 2, Fall 1996 pp. 317-334.

Maximal abelian subalgebras of the group factor of an $\tilde A_2 $ group**Authors**: Guyan Robertson (1) and Tim Steger (2)

**Author institution:**(1) Department of Mathematics, University of Newcastle, NSW 2308, AUSTRALIA

(2) Istituto di Matematica e Fisica, Universita di Sassari, via Vienna 2, 07100 Sassari, ITALIA

**Summary:**An $\tilde A_2$ group $\Gamma$ acts simply transitively on the vertices of an affine building $\Delta$. We study certain subgroups $\Gamma_0 \cong \mathbb Z^2$ which act on certain apartments of $\Delta$. 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 PukÃ¡nszky invariant contains a type $I_\infty$ summand.

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