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

Abstract:

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.

Mathematics Department, University of Newcastle, Callaghan, NSW 2308, Australia
guyan@maths.newcastle.edu.au
http://www.maths.newcastle.edu.au/~guyan/
jascki@iprimus.com.au