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

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

2004-01-26

arxiv.org/abs/gr-qc/0401101v1

J.Phys.A35:1961-1981,2002

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Latex, 27 pages, 5 figures (late submission to arxiv.org)

Scientific paper

10.1088/0305-4470/35/8/312

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.

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