Tiling systems and homology of lattices in tree products

Let $\mathrm{\Gamma }$ 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 $H_2(\mathrm{\Gamma },\text{bb}Z)$ is determined explicitly in terms of an associated 2-dimensional tiling system. It follows that under appropriate conditions the crossed product $C^{\ast }$-algebra $\text{cl}A$ associated with the action of $\mathrm{\Gamma }$ on the boundary of $\cl T_1\times \cl T_2$satisfies$\rank K_0(\cl A) = 2\cdot\rank H_2(\Gamma, \bb Z)$.