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New York Journal of Mathematics 8 (2002), 111-131.

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

Jason S. Kimberley and Guyan Robertson

Published: August 9, 2002
Keywords: group actions, trees, K-theory, C*-algebras.
Subject: Primary 20E08, 51E24; secondary 46L80


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 $T_1 \times T_2$. This article studies dynamical systems associated with BM groups. A higher rank Cuntz-Krieger algebra $\mathcal A(\G)$ is associated both with a 2-dimensional tiling system and with a boundary action of a BM group $\Gamma$. An explicit expression 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 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.

Author information:

Mathematics Department, University of Newcastle, Callaghan, NSW 2308, Australia