Möbius structures, quasi-metrics, and completeness

Abstract

We study cross ratios from an axiomatic viewpoint, also known as the study of Möbius spaces. We characterise cross ratios induced by quasi-metrics in terms of topological properties of their image. Furthermore, we generalise the notions of Cauchy-sequences and completeness to Möbius spaces and prove the existence of a unique completion under an extra assumption that, again, can be expressed in terms of the image of the cross ratio.

Revised version: The paper has been restructured and some proofs shortened. Various typos etc. corrected

References in corpus (2)

• Quasi-Mobius Homeomorphisms of Morse boundaries
• $\mathrm{CAT}(0)$ cube complexes are determined by their boundary cross ratio