Tiling systems and homology of lattices in tree products

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Let Γ be a torsion free cocompact lattice in (T₁)×(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 K₀(A) = 2· H₂(Γ, Z).