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Groups acting on products of trees, tiling systems and analytic \(K\)-theory. (English) Zbl 1025.20014

It is known that a group that acts freely and cocompactly on a tree is a finitely generated free group. However, a group that acts freely and cocompactly on a product of trees has surprising properties, this has been used by M. Burger and S. Mozes [e.g., in C. R. Acad. Sci., Paris, Sér. I 324, No. 7, 747-752 (1997; Zbl 0966.20013)] to find such properties. In this article, the authors study torsion free subgroups \(\Gamma\) of \(\operatorname{Aut}(T_1)\times\operatorname{Aut}(T_2)\), where \(T_1\), and \(T_2\) are homogeneous trees of finite degrees and that act freely and transitively of the vertex set of \(T_1\times T_2\). The authors refer to these as BM groups.
Next, associated to a BM group, via an appropriate Tits building there is a \(C^*\)-algebra \({\mathcal A}(\Gamma)\). The aim of the paper is the study of the \(K\)-theory of this \(C^*\)-algebra. Their main result is to construct an Abelian group \(C=C(\Gamma)\), completely determined by such a \(\Gamma\) that contains all the necessary ingredients, namely Proposition 5.4: Let \(\Gamma\) be a BM group then \(K_0({\mathcal A}(\Gamma))=K_1({\mathcal A}(\Gamma))=C\otimes\mathbb{Z}^{\text{rank}(C)}\). The authors also provide information about the order of the identity in \(K_0({\mathcal A}(\Gamma))\) and a complete computation of all the groups arising in this fashion when the degree of both trees is 4.

MSC:

20E08 Groups acting on trees
46L80 \(K\)-theory and operator algebras (including cyclic theory)
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
51E24 Buildings and the geometry of diagrams
05C05 Trees

Citations:

Zbl 0966.20013
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