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Tiling systems and homology of lattices in tree products

Guyan Robertson
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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 $H_2(\Gamma, \bb Z)$ is determined explicitly in terms of an associated 2-dimensional tiling system. It follows that under appropriate conditions the crossed product $C^*$-algebra $\cl A$ associated with the action of $\Gamma$ on the boundary of $\cl T_1\times \cl T_2$ satisfies $\rank K_0(\cl A) = 2\cdot\rank H_2(\Gamma, \bb Z)$.

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Guyan Robertson. 2005-11-17. Tiling systems and homology of lattices in tree products. https://arxiv.org/abs/math/0511447

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