Groups acting on products of trees, tiling systems and analytic K-theory

Let $T_1$ and $T_2$ be homogeneous trees of even degree $\ge 4$. A BM group $\mathrm{\Gamma }$ is a torsion free discrete subgroup of $\text{aut}(T_1)\times \text{aut}(T_2)$ which acts freely and transitively on the vertex set of $T_1\times T_2$. This article studies dynamical systems associated with BM groups. A higher rank Cuntz-Krieger algebra $\mathcal{A}(\text{G})$ is associated both with a 2-dimensional tiling system and with a boundary action of a BM group $\mathrm{\Gamma }$. An explicit expression is given for the K-theory of $\mathcal{A}(\text{G})$. In particular $K_0=K_1$. A complete enumeration of possible BM groups $\text{G}$ is given for a product homogeneous trees of degree 4, and the K-groups are computed.