On the inverse problem of Moebius geometry on the circle

TL;DR

This paper explores the inverse problem of identifying hyperbolic spaces from Moebius structures on the circle, introducing a canonical filling space and proving its hyperbolic properties.

Contribution

It defines a canonical 3D space called Harm for Moebius structures on the circle and proves that Harm's lines are geodesics, advancing understanding of the inverse problem.

Findings

Every line in Harm is a geodesic.

Harm is Gromov hyperbolic with the given Moebius structure.

The filling space provides a natural candidate for the inverse problem solution.

Abstract

Any (boundary continuous) hyperbolic space induces on the boundary at infinity a Moebius structure which reflects most essential asymptotic properties of the space. In this paper, we initiate the study of the inverse problem: describe Moebius structures which are induced by hyperbolic spaces at least in the simplest case of the circle. For a large class of Moebius structures on the circle, we define a canonical "filling" each of them, which serves as a natural candidate for a solution of the inverse problem. This is a 3-dimensional (pseudo)metric space Harm, which consists of harmonic 4-tuples of the respective Moebius structure with a distance determined by zig-zag paths. Our main result is the proof that every line in Harm is a geodesic, i.e., shortest in the zig-zag distance on each segment. This gives a good starting point to show that Harm is Gromov hyperbolic with the prescribed Moebius structure at infinity.