# M\"obius structures and timed causal spaces on the circle

## TL;DR

This paper explores a duality between hyperbolic spaces and spacetimes via Möbius structures on the circle, establishing a connection in 2D cases that may extend to more general settings.

## Contribution

It demonstrates how Möbius structures on the circle can generate 2D spacetimes, providing a step toward understanding the duality between hyperbolic spaces and spacetimes.

## Key findings

- Möbius structures on the circle lead to 2D spacetime models.
- A class of Möbius structures includes those from hyperbolic spaces.
- The work suggests a duality framework in 2D between hyperbolic spaces and spacetimes.

## Abstract

We discuss a conjectural duality between hyperbolic spaces on one hand and spacetimes on the other hand, living on the opposite sides of the common absolute. This duality goes via M\"obius structures on the absolute, and it is easily recognized in the classical case of symmetric rank one spaces. In a general case, no trace of such duality is known. As a first step in this direction, we show how M\"obius structures on the circle from a large class including those which stem from hyperbolic spaces give rise to 2-dimensional spacetimes, which are axiomatic versions of de Sitter 2-space, and vice versa. The paper has two Appendices, one of which is written by V.Schroeder.