Octahedral noncompact hyperbolic space forms with finite volume

Authors

  • Marica Šarac Faculty of Mining and Geology, Djušina 7, Belgrade, Yugoslavia Author

Keywords:

Noncompact hyperbolic space, regular octahedron

Subjects:

51N20, 52C22

Abstract

Following Poincarè's geometric method, we construct two new nonorientable noncompact hyperbolic space forms by the regular octahedron in Fig\.~1. The construction is motivated by Thurston's example [6], discussed also by Apansov [1] in details. Our new space forms will be denoted by D~1=H3/G1andD~2=H3/G2,where D~1 and D~2 are obtained by pairing faces of D via isometries of groups G1 and G2, respectively, acting discontinuously and freely on the hyperbolic 3-space H3 (Fig\.~2, Fig\.~3). These groups are defined by generators and relations in Sect\.~3. The complete computer classification of possible space forms by our octahedron will be discussed in [4], where it turns out that our two space forms are isometric, i.e\. G1 and G2 are conjugated by an isometry φ of H3, i.e\. G2=φ1G1φ,Misplaced &

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Published

1997-01-15