Group actions on geodesic Ptolemy spaces

TL;DR

This paper investigates geodesic Ptolemy metric spaces with specific group actions, demonstrating they are roughly isometric to Euclidean spaces, thus linking geometric group actions to classical Euclidean geometry.

Contribution

It establishes that geodesic Ptolemy spaces with proper cocompact group actions are equivariantly rough isometric to Euclidean spaces, connecting group actions with Euclidean geometry.

Findings

Spaces are equivariantly rough isometric to Euclidean spaces

Proper cocompact actions imply Euclidean-like structure

Crystallographic or virtual polycyclic groups act on these spaces

Abstract

In this paper we study geodesic Ptolemy metric spaces $X$ which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that $X$ is equivariantly rough isometric to a Euclidean space.