  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 $C^*$-algebra $C(\Omega)\rtimes\Gamma$. 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.
