Peter Kramer and Miguel Lorente 2002 J. Phys. A: Math. Gen. 35 1961 doi:10.1088/0305-4470/35/8/312
Peter Kramer1 and Miguel Lorente2
Show affiliationsThe 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.
98.80.Jk Mathematical and relativistic aspects of cosmology
85A40 Cosmology (For relativistic cosmology, see 83F05)
22E60 Lie algebras of Lie groups (For the algebraic theory of Lie algebras, see 17Bxx)
Issue 8 (1 March 2002)
Received 10 May 2001, in final form 3 January 2002
Published 15 February 2002
Peter Kramer and Miguel Lorente 2002 J. Phys. A: Math. Gen. 35 1961
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