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.

Cited by in corpus (7)

• Euler characteristics and Gysin sequences for group actions on boundaries
• New simple lattices in products of trees and their projections
• C*-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory
• The K-theory of the C*-algebras of 2-rank graphs associated to complete bipartite graphs
• Abelianization conjectures for some arithmetic square complex groups
• Centralizers in ${\tilde{A}}_2$ groups
• Tiling systems and homology of lattices in tree products