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

TL;DR

This paper computes the K-theory of crossed product C*-algebras arising from torsion-free lattices acting on $ ildeA_2$ buildings, linking geometric group actions to operator algebra invariants.

Contribution

It provides a method to compute the K-theory of these algebras, extending to a broader class of rank two Cuntz-Krieger algebras, advancing understanding of their structure.

Findings

K-theory of ${ m C}( ext{boundary}) times ext{lattice}$ computed

Extension of K-theory computation to rank two Cuntz-Krieger algebras

Classification of these algebras via K-theoretic invariants

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 $C^*$-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.