Asymptotic K-theory for groups acting on tildeA₂ buildings

Let $\Gamma$ be a torsion free lattice in $G=\PGL(3,{{\mathbb F}})$ where ${{\mathbb F}}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building ${\mathcal B}$ of $G$ 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. This article shows how to compute the K-theory of ${\mathcal A}(\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras.