Torsion in Boundary Coinvariants and K-theory for Affine Buildings

Abstract

Let $(G,{\mathfrak I},N,S)$ be an affine topological Tits system, and let $Γ$ be a torsion free cocompact lattice in $G$. This article studies the coinvariants $H_0(Γ; C(Ω,{\mathbb Z}))$, where $Ω$ is the Furstenberg boundary of $G$. It is shown that the class $[1]$ of the identity function in $H_0(Γ; C(Ω,{\mathbb Z}))$ has finite order, with explicit bounds for the order. A similar statement applies to the $K_0$ group of the boundary crossed product $C^*$-algebra $C(Ω)\rtimesΓ$. If the Tits system has type $\widetilde A_2$, exact computations are given, both for the crossed product algebra and for the reduced group $C^*$-algebra.