Asymptotic K-theory for groups acting on $\tildeA_2$ buildings

Abstract

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 $C^{\ast }$-algebra $\mathcal{A}(\Gamma )=C(\Omega )⋊\Gamma$ 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.

26 pages

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