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 $\Gamma$ be a torsion free cocompact lattice in $G$. This article studies the coinvariants $H_0(\Gamma ;C(\Omega ,\mathbb{Z}))$, where $\Omega$ is the Furstenberg boundary of $G$. It is shown that the class $[1]$ of the identity function in $H_0(\Gamma ;C(\Omega ,\mathbb{Z}))$ has finite order, with explicit bounds for the order. A similar statement applies to the $K_0$ group of the boundary crossed product $C^{\ast }$-algebra $C(\Omega )⋊\Gamma$. If the Tits system has type ${\tilde{A}}_2$, exact computations are given, both for the crossed product algebra and for the reduced group $C^{\ast }$-algebra.

References in corpus (1)

• Boundary operator algebras for free uniform tree lattices