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 H². The tessellation is analysed with tools from hyperbolic crystallography. Actions on H² 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 Φ₂ whose extension is normal in the Coxeter group. Closed geodesics arise from links on H² between octagon centres. The direction and length of the shortest closed geodesics is computed.

Dedicated to Marcos Moshinsky on the occasion of his 80th birthday.

PACS numbers: 02.40.-k, 02.20.-a, 05.50.+q, 61.43.-j, 91.10.By, 98.80.+s

Print publication: Issue 8 (1 March 2002)
Received 10 May 2001, in final form 3 January 2002
Published 15 February 2002