## 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

Zbl 0966.20013
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