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 $C^{\ast }$-algebra $\text{cA}$. 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 $\text{G}$ is a group acting freely on the vertices of an $\text{wt}{A}_2$ building, with finitely many orbits, and if $\mathrm{\Omega }$ is the boundary of that building, then $C(\text{Om})⋊\text{G}$ is the algebra associated to a certain two dimensional subshift.