Tiling systems and homology of lattices in tree products

Let Γ be a torsion-free cocompact lattice in Aut(T1) × Aut(T2), where T1, T2 are trees whose vertices all have degree at least three. The group H2(Γ,Z) is determined explicitly in terms of an associated 2-dimensional tiling system. It follows that under appropriate conditions the crossed product C∗algebra A associated with the action of Γ on the boundary of T1 ×T2 satisfies rankK0(A) = 2 · rankH2(Γ,Z).