X iv : g r - qc / 0 40 11 01 v 1 2 6 Ja n 20 04 The double torus as a 2 D cosmos : Groups , geometry , and closed geodesics
نویسندگان
چکیده
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 tesselation of its universal covering, the hyperbolic space H. The tesselation is analysed with tools from hyperbolic crystallography. Actions on H 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 H between octagon centers. The direction and length of the shortest closed geodesics is computed.
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ar X iv : g r - qc / 0 40 11 01 v 1 2 6 Ja n 20 04 The double torus as a 2 D cosmos : Groups , geometry , and closed geodesics
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 tesselation of its universal covering, the hyperbolic space H. The tesselation is analysed with tools from hyperbolic crystallography. Actions on H of groups/subgroups are identified for SU(1, 1), for a hyperbolic C...
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