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

TL;DR

This paper investigates the ergodic properties and types of boundary actions of groups acting on affine buildings of type A_n, revealing a dependence on the parity of n and classifying the actions accordingly.

Contribution

It characterizes the type of boundary actions for groups acting freely and transitively on A_n buildings, distinguishing between types tq and tqs based on n's parity.

Findings

Boundary actions are ergodic.

Action types depend on n being odd or even.

Classification into types tq and tqs.

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.