Boundary actions for affine buildings and higher rank Cuntz-Krieger algebras

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Let be a group of type rotating automorphisms of an affine building of type A₂. If acts freely on the vertices of with finitely many orbits, and if Ω is the (maximal) boundary of, then C() is a p.i.s.u.n. C*-algebra. This algebra has a structure theory analogous to that of a simple Cuntz-Krieger algebra and is the motivation for a theory of higher rank Cuntz-Krieger algebras, which has been developed by T. Steger and G. Robertson. The K-theory of these algebras can be computed explicitly in the rank two case. For the rank two examples of the form C() which arise from boundary actions on A₂ buildings, the two K-groups coincide.