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arXiv · 1812.03313

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

David Freeman·Enrico Le Donne
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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.

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David Freeman, Enrico Le Donne. 2018-12-08. Toward a quasi-M\"obius characterization of Invertible Homogeneous Metric Spaces. https://arxiv.org/abs/1812.03313

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