Möbius structures, quasimetrics and completeness (Q6596242)

Given a metric space (possibly with a point at infinity), one can mock the formula familiar from the projective line over a field in order to define a cross ratio for point quadruples. When this is done on the boundary at infinity of a CAT\((-1)\) space, then although one has many visual metrics, they all define the same cross ratio. Moreover, this cross ratio conveys information about the original CAT\((-1)\) space. This fact motivates the axiomatic study of sets equipped with a cross ratio, called Möbius structures. It is known that every cross ratio comes from a semi-metric (like a metric, but without triangle inequality). A quasimetric is a semimetric satisfying a certain weak form of triangle inequality. The author characterizes Möbius structures coming from quasimetrics. To this end, he introduces a map from point quadruples into the triangle in the real projective plane formed by the points with nonnegative homogeneous coordinates. The characteristic property is that this map stays away from the corners of the triangle.\N\NThe main results of the paper concern the topology on a Möbius structure introduced by \textit{S. Buyalo} [St. Petersbg. Math. J. 28, No. 5, 555--568 (2017; Zbl 1457.53031); translation from Algebra Anal. 28, No. 5, 1--20 (2017)] and a notion of Cauchy sequences defined by the author for Möbius structures coming from a quasimetric with a certain symmetry property. Namely, for Möbius structures defined by a metric, the metric topology and the topology coming from the Möbius structure agree. The Möbius structure defined by a (possibly extended) metric space is complete if and only if the metric space is complete and is either bounded or has a point at infinity. For Möbius structures induced by quasimetrics with the symmetry property, the author finally constructs a completion that enjoys the appropriate universal property.