The double torus as a 2D cosmos: groups, geometry and closed geodesics

TL;DR

This paper models a 2D universe with double torus topology using hyperbolic geometry, analyzing its tessellation, symmetry groups, and closed geodesics to understand its geometric and topological properties.

Contribution

It introduces a hyperbolic tessellation approach to model a 2D cosmos with genus 2 topology, identifying symmetry groups and calculating shortest closed geodesics.

Findings

Identified group actions on hyperbolic space relevant to the double torus.

Analyzed the octagonal tessellation and its symmetry properties.

Computed the direction and length of shortest closed geodesics.

Abstract

The double torus provides a relativistic model for a closed 2D cosmos with topology of genus 2 and constant negative curvature. Its unfolding into an octagon extends to an octagonal tessellation of its universal covering, the hyperbolic space H^2. The tessellation is analysed with tools from hyperbolic crystallography. Actions on H^2 of groups/subgroups are identified for SU(1, 1), for a hyperbolic Coxeter group acting also on SU(1, 1), and for the homotopy group \Phi_2 whose extension is normal in the Coxeter group. Closed geodesics arise from links on H^2 between octagon centres. The direction and length of the shortest closed geodesics is computed.