Let T₁ and T₂ be homogeneous trees of even degree ≧ 4. A BM group Γ is a torsion-free discrete subgroup of Aut(T₁)×Aut(T₂) which acts freely and transitively on the vertex set of T₁×T₂. This article studies dynamical systems associated with BM groups. A higher rank Cuntz-Krieger algebra A(Γ) is associated both with a 2-dimensional tiling system and with a boundary action of a BM group Γ. An explicit expression is given for the K-theory of A(Γ). In particular K₀=K₁. A complete enumeration of possible BM groups Γ is given for a product homogeneous trees of degree 4, and the K-groups are computed.

This research was funded by the Australian Research Council. The second author is also grateful for the support of the University of Geneva.