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Let $T_1$ and $T_2$ be homogeneous trees of even degree $\ge 4$. A BM group$\Gamma$ is a torsion free discrete subgroup of $\aut (T_1) \times \aut (T_2)$which acts freely and transitively on the vertex set of $T_1 \times T_2$. Thisarticle studies dynamical systems associated with BM groups. A higher rankCuntz-Krieger algebra $\mathcal A(\G)$ is associated both with a 2-dimensionaltiling system and with a boundary action of a BM group $\Gamma$. An explicitexpression is given for the K-theory of $\mathcal A(\G)$. In particular$K_0=K_1$. A complete enumeration of possible BM groups $\G$ is given for aproduct homogeneous trees of degree 4, and the K-groups are computed.

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