Abstract
We identify the two minimal co-volume lattices of the isometry group of hyperbolic $3$-space that contain a finite spherical triangle group. These two groups are arithmetic and are in fact the two minimal co-volume lattices. Our results here represent the key step in establishing this fact, thereby solving a problem posed by Siegel in 1945. As a consequence we obtain sharp bounds on the order of the symmetry group of a hyperbolic $3$-manifold in terms of its volume, analogous to the Hurwitz $84g-84$ theorem of 1892.
The finite spherical subgroups of a Kleinian group give rise to the vertices of the singular graph in the quotient orbifold. We identify the small values of the discrete spectrum of hyperbolic distances between these vertices and show these small values give rise to arithmetic lattices. Once vertices are sufficiently separated, one obtains volume bounds by studying equivariant sets.
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AUTHOR = {Agol, Ian and Storm, Peter A. and Thurston, William P.},
TITLE = {Lower bounds on volumes of hyperbolic {H}aken 3-manifolds},
INVNOTE = {With an appendix by Nathan Dunfield},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {20},
YEAR = {2007},
NUMBER = {4},
PAGES = {1053--1077},
ISSN = {0894-0347},
MRCLASS = {53C44 (57M50 57N10)},
MRNUMBER = {2008i:53086},
MRREVIEWER = {Colin C. Adams},
DOI = {10.1090/S0894-0347-07-00564-4},
} -
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AUTHOR = {Gehring, F. W. and Martin, G. J.},
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AUTHOR = {Margulis, G. A.},
TITLE = {Discrete groups of motions of manifolds of nonpositive curvature},
BOOKTITLE = {Proceedings of the International Congress of Mathematicians, II},
VENUE = {Vancouver, B.C., 1974},
PAGES = {21--34},
PUBLISHER = {Canad. Math. Congress, Montreal, Que.},
marginpar = {check output},
YEAR = {1975},
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MRNUMBER = {58 \#11226},
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MRREVIEWER = {J. S. Joel},
} -
[MM] T. H. Marshall and G. J. Martin, "Minimal co-volume hyperbolic lattices, II: Simple torsion in Kleinian groups," , preprint , 2008.
@techreport{MM,
author = {Marshall, T. H. and Martin, G. J.},
TITLE = {Minimal co-volume hyperbolic lattices, {II}: Simple torsion in {K}leinian groups},
TYPE = {preprint},
YEAR = {2008},
invmarginpar = {where can this be found?},
} -
[Mart] G. J. Martin, "On the geometry of Kleinian groups," in Quasiconformal Mappings and Analysis, Duren, P., Heinonen, J., Osgood, B., and Palka, B., Eds., New York: Springer-Verlag, 1998, pp. 253-274.
@incollection{Mart, MRKEY = {1488454},
AUTHOR = {Martin, Gaven J.},
EDITOR = {Duren, Peter and Heinonen, Juha and Osgood, Brad and Palka, Bruce},
TITLE = {On the geometry of {K}leinian groups},
BOOKTITLE = {Quasiconformal Mappings and Analysis},
VENUE = {Ann Arbor, MI, 1995},
PAGES = {253--274},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1998},
MRCLASS = {30F40},
MRNUMBER = {98k:30061},
ZBLNUMBER = {0886.30030},
MRREVIEWER = {I. Kra},
} -
[Mart2] G. J. Martin, "The volume of regular tetrahedra and sphere packing in hyperbolic $3$-space," Mathematical Chronicle, vol. 20, pp. 127-147, 1991.
@article{Mart2, MRKEY = {1137879},
AUTHOR = {Martin, Gaven J.},
TITLE = {The volume of regular tetrahedra and sphere packing in hyperbolic {$3$}-space},
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JOURNAL = {Mathematical Chronicle},
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YEAR = {1991},
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MRNUMBER = {93d:52025},
ZBLNUMBER = {0761.52012},
MRREVIEWER = {A. Florian},
} -
[M] B. Maskit, Kleinian Groups, New York: Springer-Verlag, 1988.
@book{M, MRKEY = {959135},
AUTHOR = {Maskit, Bernard},
TITLE = {Kleinian {G}roups},
SERIES = {Grundl. Math. Wissen.},
NUMBER = {287},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1988},
PAGES = {xiv+326},
ISBN = {3-540-17746-9},
MRCLASS = {30F40 (20H10 22E40)},
MRNUMBER = {90a:30132},
ZBLNUMBER = {0627.30039},
MRREVIEWER = {William Abikoff},
} -
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R. Meyerhoff, "Sphere-packing and volume in hyperbolic $3$-space," Comment. Math. Helv., vol. 61, iss. 2, pp. 271-278, 1986.
@article{Mey, MRKEY = {856090},
AUTHOR = {Meyerhoff, Robert},
TITLE = {Sphere-packing and volume in hyperbolic {$3$}-space},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {61},
YEAR = {1986},
NUMBER = {2},
PAGES = {271--278},
ISSN = {0010-2571},
CODEN = {COMHAX},
MRCLASS = {52A45 (52A40)},
MRNUMBER = {88e:52023},
ZBLNUMBER = {0611.57010},
MRREVIEWER = {A. Florian},
DOI = {10.1007/BF02621915},
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G. D. Mostow, "Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms," Inst. Hautes Études Sci. Publ. Math., vol. 34, pp. 53-104, 1968.
@article{Mostow, MRKEY = {0236383},
AUTHOR = {Mostow, G. D.},
TITLE = {Quasi-conformal mappings in {$n$}-space and the rigidity of hyperbolic space forms},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {34},
YEAR = {1968},
PAGES = {53--104},
ISSN = {0073-8301},
MRCLASS = {30.47},
MRNUMBER = {38 \#4679},
ZBLNUMBER = {0189.09402},
MRREVIEWER = {R. J. Cannon},
URL = {http://www.numdam.org/item?id=PMIHES_1968__34__53_0},
} -
[Rat] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, New York: Springer-Verlag, 1994.
@book{Rat, MRKEY = {1299730},
AUTHOR = {Ratcliffe, John G.},
TITLE = {Foundations of {H}yperbolic {M}anifolds},
SERIES = {Grad. Texts Math.},
NUMBER = {149},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1994},
PAGES = {xii+747},
ISBN = {0-387-94348-X},
MRCLASS = {57M50 (20H10 30F40 51M10)},
MRNUMBER = {95j:57011},
ZBLNUMBER = {0809.51001},
MRREVIEWER = {Colin C. Adams},
} -
[Selberg] A. Selberg, "On discontinuous groups in higher-dimensional symmetric spaces," in Contributions to Function Theory, Bombay: Tata Inst. Fundamental Res., 1960, pp. 147-164.
@incollection{Selberg, MRKEY = {0130324},
AUTHOR = {Selberg, Atle},
TITLE = {On discontinuous groups in higher-dimensional symmetric spaces},
FBOOKTITLE = {Contributions to Function theory (internat. Colloq. Function Theory, Bombay, 1960)},
BOOKTITLE = {Contributions to Function Theory},
VENUE = {Bombay, 1960},
PAGES = {147--164},
PUBLISHER = {Tata Inst. Fundamental Res.},
FPUBLISHER = {Tata Institute of Fundamental Research},
ADDRESS = {Bombay},
YEAR = {1960},
MRCLASS = {20.65 (22.70)},
MRNUMBER = {24 \#A188},
ZBLNUMBER = {0201.36603},
MRREVIEWER = {E. Grosswald},
} -
@article{Siegel, MRKEY = {0009959},
AUTHOR = {Siegel, Carl Ludwig},
TITLE = {Discontinuous groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {44},
YEAR = {1943},
PAGES = {674--689},
ISSN = {0003-486X},
MRCLASS = {20.0X},
MRNUMBER = {5,228e},
ZBLNUMBER = {0061.04504},
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DOI = {10.2307/1969104},
} -
[Siegel2]
C. L. Siegel, "Some remarks on discontinuous groups," Ann. of Math., vol. 46, pp. 708-718, 1945.
@article{Siegel2, MRKEY = {0014088},
AUTHOR = {Siegel, Carl Ludwig},
TITLE = {Some remarks on discontinuous groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {46},
YEAR = {1945},
PAGES = {708--718},
ISSN = {0003-486X},
MRCLASS = {20.0X},
MRNUMBER = {7,239c},
ZBLNUMBER = {0061.04505},
MRREVIEWER = {S. A. Jennings},
DOI = {10.2307/1969206},
} -
[V] È. B. Vinberg, "Hyperbolic groups of reflections," Uspekhi Mat. Nauk, vol. 40, pp. 29-66, 255, 1985.
@article{V, MRKEY = {783604},
AUTHOR = {Vinberg, {È}. B.},
TITLE = {Hyperbolic groups of reflections},
JOURNAL = {Uspekhi Mat. Nauk},
NOTE = {In Russian, translated in {\it Russian Math. Surveys} {\bf 40} (1985), 31--75},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
VOLUME = {40},
YEAR = {1985},
INVNUMBER = {1(241)},
PAGES = {29--66, 255},
ISSN = {0042-1316},
MRCLASS = {53C20 (20H15 22E40 32M10)},
MRNUMBER = {86m:53059},
ZBLNUMBER = {0579.51015},
MRREVIEWER = {L. N. Vaserstein},
} -
[Y]
A. Yamada, "On Marden’s universal constant of Fuchsian groups," Kodai Math. J., vol. 4, iss. 2, pp. 266-277, 1981.
@article{Y, MRKEY = {630246},
AUTHOR = {Yamada, Akira},
TITLE = {On {M}arden's universal constant of {F}uchsian groups},
JOURNAL = {Kodai Math. J.},
FJOURNAL = {Kodai Mathematical Journal},
VOLUME = {4},
YEAR = {1981},
NUMBER = {2},
PAGES = {266--277},
ISSN = {0386-5991},
MRCLASS = {30F35},
MRNUMBER = {83k:30047},
ZBLNUMBER = {0469.30038},
MRREVIEWER = {L. Keen},
URL = {http://projecteuclid.org/getRecord?id=euclid.kmj/1138036373},
}