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

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 $Ω$ is the boundary of that building, then $C(\Om)\rtimes \G$ is the algebra associated to a certain two dimensional subshift.

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