Asymptotic \(K\)-theory for groups acting on \(\widetilde A_2\) buildings (Q2762713)

Let \({\mathbb F}\) be a nonarchimedean local field and let \(\Gamma\) be a torsion free lattice in \(PGL(3,{\mathbb F})\). Then \(\Gamma\) acts freely on the affine Bruhat-Tits building \({\mathcal B}\) of \(PGL(3,{\mathbb F})\) and there is an induced action on the boundary \(\Omega\) of \({\mathcal B}\). The crossed product \(C^*\)-algebra \({\mathcal A}(\Gamma)=C(\Omega)\rtimes\Gamma\) depends only on \(\Gamma\) and is classified by its \(K\)-theory. It is shown in the paper, how to compute the \(K\)-theory of \({\mathcal A}(\Gamma)\) and of the larger class of rank two Cuntz-Krieger \(C^*\)-algebras.