Type III actions on boundaries of $\tilde A_n$ buildings

Abstract

Let $\Gamma$ be a group of type rotating automorphisms of a building $\fX$ of type $\tilde A_n$ and order $q$. Suppose that $\G$ acts freely and transitively on the vertex set of $\fX$. Then the action of $\Gamma$ on the boundary of $\fX$ is ergodic, of type $\tq$ or type $\tqs$ depending on whether $n$ is odd or even.