Volumes of hyperbolic L\"obell 3-manifolds
Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 17-23.

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In 1931 F. Löbell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing Löbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.
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A. Yu. Vesnin. Volumes of hyperbolic L\"obell 3-manifolds. Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 17-23. http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a2/

[1] Vinberg E. B., Shvartsman O. V., “Diskretnye gruppy dvizhenii prostranstv postoyannoi krivizny”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundament. napravleniya, 29, VINITI, M., 1988, 147–259

[2] Volf Dzh., Prostranstva postoyannoi krivizny, Nauka, M., 1982

[3] Klein F., Neevklidova geometriya, ONTI NKTP, M., 1936

[4] Löbell F., “Beispiele geschlossener dreidimensionaler Clifford–Kleinischer Räume negativer Krümmung”, Ber. Verh. Sächs. Akad. Leipzig. Math.-Phys. Kl., 83 (1931), 168–174

[5] Vesnin A. Yu., “Trekhmernye giperbolicheskie mnogoobraziya tipa Lëbellya”, Sib. matem. zh., 28:5 (1987), 50–53 | MR

[6] Mednykh A. D., Vesnin A. Yu., “On three-dimensional hyperbolic manifolds of Löbell type”, Complex Analysis and Applications'85, Sofia, 1986, 440–446 | MR | Zbl

[7] Mednykh A. D., “Gruppy avtomorfizmov trekhmernykh giperbolicheskikh mnogoobrazii”, Dokl. AN SSSR, 285:1 (1985), 40–44 | MR | Zbl

[8] Vesnin A. Yu., “Trekhmernye giperbolicheskie mnogoobraziya s obschim fundamentalnym mnogogrannikom”, Matem. zametki, 49:6 (1991), 29–32 | MR | Zbl

[9] Gromov M., “Hyperbolic manifolds according to Thurston and Jørgensen”, Lecture Notes in Math., 842, Springer, Berlin, 1981, 40–53 | MR

[10] Apanasov B. N., Gutsul I. S., “Greatly symmetric totally geodesic surfaces and closed hyperbolic 3-manifolds which share a fundamental polyhedron”, Topology'90, eds. B. Apanasov, W. Neumann, A. Ried, L. Siebenmann, de Gruyter, Berlin, 1992, 37–53 | MR

[11] Zimmermann B., “A note on hyperbolic 3-manifolds of the same volume”, Monatsh. Math., 117 (1994), 139–143 | DOI | MR | Zbl

[12] Weeks J., SnapPea. Version 5/18/92, A program for the Macintosh to compute hyperbolic structures on 3-manifolds

[13] Alekseevskii D. V., Vinberg E. B., Solodovnikov A. S., “Geometriya prostranstv postoyannoi krivizny”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundament. napravleniya, 29, VINITI, M., 1988, 5–146

[14] Kellerhals R., “On the volume of hyperbolic polyhedra”, Math. Ann., 285 (1989), 541–569 | DOI | MR | Zbl

[15] Milnor J., “Hyperbolic geometry: the first 150 years”, Bull. Amer. Math. Soc., 6 (1982), 9–24 | DOI | MR | Zbl

[16] Andreev E. M., “O vypuklykh mnogogrannikakh v prostranstvakh Lobachevskogo”, Matem. sb., 81(123):3 (1970), 445–478 | MR | Zbl

[17] Al-Jubouri N. K., “On nonorientable hyperbolic 3-manifolds”, Chinese Quart. J. Math., 31 (1980), 9–18 | MR | Zbl