Tiling systems and homology of lattices in tree products (Q2581086)

Consider the two-dimensional cell complex \(\Delta=T_1\times T_2\), where \(T_1\), \(T_2\) are locally finite trees whose vertices have degree at least three and let \(\Gamma\) be a discrete subgroup of \(\text{Aut}(T_1)\times\text{Aut}(T_2)\) which acts freely and cocompactly on \(\Delta\). Associated to this action is a tiling system whose set of tiles is the set \(R\) of `directed' 2-cells of \(\Gamma\setminus\Delta\). The author constructs a map \(\mathbb ZR\to\mathbb ZR\oplus\mathbb ZR\) and shows that \(H_2(\Gamma,\mathbb Z)\) coincides with the kernel of that map. If \(p\) is prime then PGL\(_2(\mathbb Q_p)\) acts on its Bruhat--Tits tree \(T_{p+1}\), which is a homogeneous tree of degree \(p+1\). If \(p,l\) are prime then the group \(G=\text{PGL}_2(\mathbb Q_p)\times \text{PGL}_2(\mathbb Q_l)\) acts on \(\Delta=T_{p+1}\times T_{l+1}\). Let \(\Gamma\) be a torsion-free irreducible lattice in \(G\). Then the crossed-product \(C^*\)-algebra \(A(\Gamma,\partial\Delta)=C(\partial\Delta)\rtimes\Gamma\) obtained from the action of \(\Gamma\) on the boundary \(\partial\Delta\) is a higher rank Cuntz-Krieger algebra and can be classified by \(K\)-theory. As a corollary of the main result, the author shows that \({ rank\,}K_0(A(\Gamma,\partial\Delta))=2\cdot {rank\,}H_2(\Gamma,\mathbb Z)\). Since \(H_1(\Gamma,\mathbb Z)\) is finite, it is calculated that \(\chi(\Gamma)-\frac{(p-1)(q-1)}{4}| X^0| \), where \(| X^0| \) is the number of vertices of \(X=\Gamma\setminus\Delta\).