Tiling systems and homology of lattices in tree products

Abstract

Let $\Gamma$ be a torsion free cocompact lattice in $\aut(\cl T_1)\times\aut(\cl T_2)$, where $\cl T_1$, $\cl T_2$ are trees whose vertices all have degree at least three. The group $H_2(Γ, \bb Z)$ is determined explicitly in terms of an associated 2-dimensional tiling system. It follows that under appropriate conditions the crossed product $C^{\ast }$-algebra $\cl A$ associated with the action of $\Gamma$ on the boundary of $\cl T_1\times \cl T_2$ satisfies $\rank K_0(\cl A) = 2\cdot\rank H_2(Γ, \bb Z)$.

arXiv comment12 pages

References in corpus (1)

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