# Inverse problem for Moebius geometry on the circle

## TL;DR

This paper solves the inverse problem in Moebius geometry on the circle by characterizing a class of structures that correspond to hyperbolic space boundaries, showing this class is non-empty and topologically open.

## Contribution

It provides a description of a class of Moebius structures on the circle that are realizable as boundaries of hyperbolic spaces, expanding understanding of boundary structures.

## Key findings

- Identifies a non-empty class of Moebius structures on the circle.
- Shows this class forms an open neighborhood of the canonical structure.
- Establishes a link between Moebius structures and hyperbolic space boundaries.

## Abstract

We give a solution to the inverse problem of Moebius geometry on the circle. Namely, we describe a class of Moebius structures on the circle for each of which there is a hyperbolic space such that its boundary at infinity is the circle, and the induced Moebius structure coincides with the given one. That class is not empty and form an open neighborhood of the canonical Moebius structure in an appropriate fine topology.