  Let $\Gamma$ be a torsion free cocompact lattice in $\aut(\cl
T_1)\times\aut(\cl T_2)$, where $\cl T_1$, $\cl T_2$ are trees whose vertices
all have degree at least three. The group $H_2(\Gamma, \bb Z)$ is determined
explicitly in terms of an associated 2-dimensional tiling system. It follows
that under appropriate conditions the crossed product $C^*$-algebra $\cl A$
associated with the action of $\Gamma$ on the boundary of $\cl T_1\times \cl
T_2$ satisfies $\rank K_0(\cl A) = 2\cdot\rank H_2(\Gamma, \bb Z)$.

K-Theory and Homology;Operator Algebras