Dynamical boundaries of affine buildings: C*-simplicity and Poisson boundaries

TL;DR

This paper studies groups acting on affine buildings, proving C*-simplicity and identifying Poisson boundaries, by analyzing their dynamics through flag limit sets and constructing equivariant barycenter maps.

Contribution

It introduces the class of groups of general type acting on affine buildings, proving C*-simplicity and characterizing their Poisson boundaries with new dynamical methods.

Findings

Groups of general type are C*-simple due to topologically free flag limit sets.

Flag limit sets can be mean proximal, carrying unique stationary measures.

Lattices in affine buildings serve as examples with identified Poisson boundaries.

Abstract

We investigate a class of groups acting on possibly exotic affine buildings $X$ and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of chambers at infinity of $X$. For a group $G$ of general type, we prove C*-simplicity by showing that its flag limit set $\Lambda_{\mathcal F}(G)$ is topologically free, minimal, and strongly proximal. When $\Lambda_{\mathcal F}(G)$ intersects all Schubert cells relative to a limit chamber, then it is a mean proximal space, in the sense that it carries a unique proximal stationary measure for any admissible probability measure on the acting group. Lattices are established as examples of groups of general type, and their Poisson boundaries are identified. The arguments rely on constructing an equivariant barycenter map from triples of chambers in generic position to the affine building.