# Toward a quasi-M\"obius characterization of Invertible Homogeneous Metric Spaces

## TL;DR

This paper explores the properties of locally compact metric spaces with M"obius homogeneity, providing new characterizations of boundaries of rank-one symmetric spaces and analyzing the implications of various homogeneity conditions.

## Contribution

It introduces a novel characterization of snowflakes of boundaries of rank-one symmetric spaces and links homogeneity with bi-Lipschitz properties and quasi-invertibility.

## Key findings

- Characterization of snowflakes of boundaries of rank-one symmetric spaces.
- Connections between M"obius homogeneity and bi-Lipschitz homogeneity.
- Metric properties of spaces with and without cut points.

## Abstract

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to M\"obius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-M\"obius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.