Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras

TL;DR

This paper introduces higher rank Cuntz-Krieger algebras associated with certain subshifts of finite type, exploring their properties and a specific example involving group actions on buildings.

Contribution

It constructs a new class of higher rank Cuntz-Krieger algebras from subshifts and analyzes their properties, including a concrete example involving group actions on buildings.

Findings

The algebra is simple, purely infinite, and nuclear.

A specific example relates group actions on buildings to subshift C*-algebras.

The construction generalizes classical Cuntz-Krieger algebras to higher dimensions.

Abstract

To an $r$-dimensional subshift of finite type satisfying certain special properties we associate a $C^*$-algebra $\cA$. This algebra is a higher rank version of a Cuntz-Krieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if $\G$ is a group acting freely on the vertices of an $\wt A_2$ building, with finitely many orbits, and if $\Omega$ is the boundary of that building, then $C(\Om)\rtimes \G$ is the algebra associated to a certain two dimensional subshift.