Torsion in K-theory for boundary actions on affine buildings of type $\tA_n$

Abstract

Let $\Gamma$ be a torsion free lattice in $G = \PGL(n+1,\FF)$, where $n\ge 1$ and $\FF$ is a non-archimedean local field. Then $\Gamma$ acts on the Furstenberg boundary $G/P$, where $P$ is a minimal parabolic subgroup of $G$. The identity element $\id$ in the crossed product $C^*$-algebra $C(G/P)\rtimes \Gamma$ generates a class $[\id]$ in the $K_0$ group of $C(G/P)\rtimes \Gamma$. It is shown that $[\id]$ is a torsion element of $K_0$ and there is an explicit bound for the order of $[\id]$. The result is proved more generally for groups acting on affine buildings of type $\tA_n$. For $n=1, 2$ the Euler-Poincar\'e characteristic $\chi(\Gamma)$ annihilates the class $[\id]$.