Some remarks on proper actions, proper metric spaces, and buildings

TL;DR

This paper explores isometric group actions on proper metric spaces, demonstrating that certain transitive actions on Euclidean buildings imply strong transitivity on their maximal atlas, with implications for geometric group theory.

Contribution

It establishes that proper and Weyl transitive actions on Euclidean buildings are strongly transitive on the complete apartment system, linking group actions to building structure.

Findings

Proper and Weyl transitive actions imply strong transitivity on the building's atlas

Connections between group actions and geometric structures of buildings

Insights into isometric actions on proper metric spaces

Abstract

We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.