Abstract
This paper represents the final step in solving the problem, posed by Siegel in 1945, of determining the minimal co-volume lattices of hyperbolic $3$-space $\mathbb{H}$ (also Problem 3.60 (F) in the Kirby problem list from 1993). Here we identify the two smallest co-volume lattices. Both these groups are two-generator arithmetic lattices, generated by two elements of finite orders 2 and 3. Their co-volumes are $0.0390\dots$ and $ 0.0408\dots$; the precise values are given in terms of the Dedekind zeta function of a number field via a formula of Borel.
Our earlier work dealt with the cases when there is a finite spherical subgroup or high order torsion in the lattice. Thus, here we are concerned with the study of simple torsion of low order and the geometric structure of Klein 4-subgroups of a Kleinian group. We also identify certain universal geometric constraints imposed by discreteness on Kleinian groups which are of independent interest.
To obtain these results we use a range of geometric and arithmetic criteria to obtain information on the structure of the singular set of the associated orbifold and then co-volume bounds by studying equivariant neighbourhoods of fixed point sets, together with a rigorous computer search of certain parameter spaces for two-generator Kleinian groups.
-
[A] B. N. Apanasov, "A certain universal property of Kleinian groups in a hyperbolic metric," Dokl. Akad. Nauk SSSR, vol. 225, iss. 1, pp. 15-18, 1975.
@article {A, MRKEY = {0402034},
AUTHOR = {Apanasov, B. N.},
TITLE = {A certain universal property of {K}leinian groups in a hyperbolic metric},
JOURNAL = {Dokl. Akad. Nauk SSSR},
FJOURNAL = {Doklady Akademii Nauk SSSR},
VOLUME = {225},
YEAR = {1975},
NUMBER = {1},
PAGES = {15--18},
ISSN = {0002-3264},
MRCLASS = {30A58},
MRNUMBER = {0402034},
MRREVIEWER = {O. M. Fomenko},
} -
[An] E. M. Andreev, "Convex polyhedra in Lobačevskiĭspaces," Mat. Sb., vol. 81 (123), pp. 445-478, 1970.
@article {An, MRKEY = {0259734},
AUTHOR = {Andreev, E. M.},
TITLE = {Convex polyhedra in {L}obačevskiĭspaces},
JOURNAL = {Mat. Sb.},
VOLUME = {81 (123)},
YEAR = {1970},
PAGES = {445--478},
MRCLASS = {50.40 (52.00)},
MRNUMBER = {0259734},
MRREVIEWER = {M. Decuyper},
} -
[B] A. F. Beardon, The Geometry of Discrete Groups, New York: Springer-Verlag, 1983, vol. 91.
@book {B, MRKEY = {0698777},
AUTHOR = {Beardon, Alan F.},
TITLE = {The Geometry of Discrete Groups},
SERIES = {Grad. Texts in Math.},
VOLUME = {91},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1983},
PAGES = {xii+337},
ISBN = {0-387-90788-2},
MRCLASS = {22E40 (11F06 20H15 30F35 57N10)},
MRNUMBER = {0698777},
MRREVIEWER = {Norbert Wielenberg},
ZBLNUMBER = {0528.30001},
} -
[Bor]
A. Borel, "Commensurability classes and volumes of hyperbolic $3$-manifolds," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 8, iss. 1, pp. 1-33, 1981.
@article {Bor, MRKEY = {0616899},
AUTHOR = {Borel, A.},
TITLE = {Commensurability classes and volumes of hyperbolic {$3$}-manifolds},
JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV},
VOLUME = {8},
YEAR = {1981},
NUMBER = {1},
PAGES = {1--33},
CODEN = {PSNAAI},
MRCLASS = {22E40 (20G30 51M25 57N10)},
MRNUMBER = {0616899},
MRREVIEWER = {Avner Ash},
URL = {http://www.numdam.org/item?id=ASNSP_1981_4_8_1_1_0},
ZBLNUMBER = {0473.57003},
} -
[Bo]
K. Böröczky, "Packing of spheres in spaces of constant curvature," Acta Math. Acad. Sci. Hungar., vol. 32, iss. 3-4, pp. 243-261, 1978.
@article {Bo, MRKEY = {0512399},
AUTHOR = {B{ö}r{ö}czky, K.},
TITLE = {Packing of spheres in spaces of constant curvature},
JOURNAL = {Acta Math. Acad. Sci. Hungar.},
FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},
VOLUME = {32},
YEAR = {1978},
NUMBER = {3-4},
PAGES = {243--261},
ISSN = {0001-5954},
CODEN = {ACMTAV},
MRCLASS = {52A45 (52A40)},
MRNUMBER = {0512399},
MRREVIEWER = {A. Florian},
DOI = {10.1007/BF01902361},
ZBLNUMBER = {0422.52011},
} -
[Cao] C. Cao, "On three generator Möbius groups," New Zealand J. Math., vol. 23, iss. 2, pp. 111-120, 1994.
@article {Cao, MRKEY = {1313446},
AUTHOR = {Cao, Chun},
TITLE = {On three generator {M}öbius groups},
JOURNAL = {New Zealand J. Math.},
FJOURNAL = {New Zealand Journal of Mathematics},
VOLUME = {23},
YEAR = {1994},
NUMBER = {2},
PAGES = {111--120},
ISSN = {1171-6096},
MRCLASS = {30F40},
MRNUMBER = {1313446},
MRREVIEWER = {Shigeyasu Kamiya},
ZBLNUMBER = {0839.30048},
} -
[Cao2]
C. Cao, "Some trace inequalities for discrete groups of Möbius transformations," Proc. Amer. Math. Soc., vol. 123, iss. 12, pp. 3807-3815, 1995.
@article {Cao2, MRKEY = {1301013},
AUTHOR = {Cao, Chun},
TITLE = {Some trace inequalities for discrete groups of {M}öbius transformations},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {123},
YEAR = {1995},
NUMBER = {12},
PAGES = {3807--3815},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {30F40},
MRNUMBER = {1301013},
MRREVIEWER = {Grzegorz Gromadzki},
DOI = {10.2307/2161910},
} -
[CGM] C. Cao, F. W. Gehring, and G. J. Martin, "Lattice constants and a lemma of Zagier," in Lipa’s Legacy, Providence, RI: Amer. Math. Soc., 1997, vol. 211, pp. 107-120.
@incollection {CGM, MRKEY = {1476983},
AUTHOR = {Cao, C. and Gehring, F. W. and Martin, G. J.},
TITLE = {Lattice constants and a lemma of {Z}agier},
BOOKTITLE = {Lipa's Legacy},
VENUE={{N}ew {Y}ork, 1995},
SERIES = {Contemp. Math.},
VOLUME = {211},
PAGES = {107--120},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1997},
MRCLASS = {30F40 (11H06 20H10 57S30)},
MRNUMBER = {1476983},
MRREVIEWER = {C. Maclachlan},
ZBLNUMBER = {0884.30034},
} -
[CG] L. Carleson and T. W. Gamelin, Complex Dynamics, New York: Springer-Verlag, 1993.
@book {CG, MRKEY = {1230383},
AUTHOR = {Carleson, Lennart and Gamelin, Theodore W.},
TITLE = {Complex Dynamics},
SERIES = {Universitext: Tracts in Math.},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {x+175},
ISBN = {0-387-97942-5},
MRCLASS = {30D05 (30C62 39B12 58F23)},
MRNUMBER = {1230383},
MRREVIEWER = {Ben Bielefeld},
ZBLNUMBER = {0782.30022},
} -
[ChinFre]
T. Chinburg and E. Friedman, "The smallest arithmetic hyperbolic three-orbifold," Invent. Math., vol. 86, iss. 3, pp. 507-527, 1986.
@article {ChinFre, MRKEY = {0860679},
AUTHOR = {Chinburg, Ted and Friedman, Eduardo},
TITLE = {The smallest arithmetic hyperbolic three-orbifold},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {86},
YEAR = {1986},
NUMBER = {3},
PAGES = {507--527},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {22E40 (57N10)},
MRNUMBER = {0860679},
MRREVIEWER = {O. V. Shvartsman},
DOI = {10.1007/BF01389265},
ZBLNUMBER = {0643.57011},
} -
[CMT] M. Conder, G. Martin, and A. Torstensson, "Maximal symmetry groups of hyperbolic 3-manifolds," New Zealand J. Math., vol. 35, iss. 1, pp. 37-62, 2006.
@article {CMT, MRKEY = {2222175},
AUTHOR = {Conder, Marston and Martin, Gaven and Torstensson, Anna},
TITLE = {Maximal symmetry groups of hyperbolic 3-manifolds},
JOURNAL = {New Zealand J. Math.},
FJOURNAL = {New Zealand Journal of Mathematics},
VOLUME = {35},
YEAR = {2006},
NUMBER = {1},
PAGES = {37--62},
ISSN = {1171-6096},
MRCLASS = {57M60 (20B25 57M50 57N10)},
MRNUMBER = {2222175},
MRREVIEWER = {Athanase Papadopoulos},
ZBLNUMBER = {1104.20035},
} -
[CS]
M. Culler and P. B. Shalen, "Paradoxical decompositions, $2$-generator Kleinian groups, and volumes of hyperbolic $3$-manifolds," J. Amer. Math. Soc., vol. 5, iss. 2, pp. 231-288, 1992.
@article {CS, MRKEY = {1135928},
AUTHOR = {Culler, Marc and Shalen, Peter B.},
TITLE = {Paradoxical decompositions, {$2$}-generator {K}leinian groups, and volumes of hyperbolic {$3$}-manifolds},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {5},
YEAR = {1992},
NUMBER = {2},
PAGES = {231--288},
ISSN = {0894-0347},
MRCLASS = {57M50 (30F40)},
MRNUMBER = {1135928},
MRREVIEWER = {Colin C. Adams},
DOI = {10.2307/2152768},
ZBLNUMBER = {0769.57010},
} -
[DM] D. A. Derevnin and A. D. Mednykh, "Geometric properties of discrete groups acting with fixed points in a Lobachevskiĭspace," Dokl. Akad. Nauk SSSR, vol. 300, iss. 1, pp. 27-30, 1988.
@article {DM, MRKEY = {0948799},
AUTHOR = {Derevnin, D. A. and Mednykh, A. D.},
TITLE = {Geometric properties of discrete groups acting with fixed points in a {L}obachevskiĭspace},
JOURNAL = {Dokl. Akad. Nauk SSSR},
FJOURNAL = {Doklady Akademii Nauk SSSR},
VOLUME = {300},
YEAR = {1988},
NUMBER = {1},
PAGES = {27--30},
ISSN = {0002-3264},
MRCLASS = {30F40 (11F06 22E40 30F35 57S20)},
MRNUMBER = {0948799},
MRREVIEWER = {N. A. Gusevski{\u\i}},
ZBLNUMBER = {0713.30044},
} -
[GaMM]
D. Gabai, R. G. Meyerhoff, and P. Milley, "Volumes of tubes in hyperbolic 3-manifolds," J. Differential Geom., vol. 57, iss. 1, pp. 23-46, 2001.
@article {GaMM, MRKEY = {1871490},
AUTHOR = {Gabai, David and Meyerhoff, G. Robert and Milley, Peter},
TITLE = {Volumes of tubes in hyperbolic 3-manifolds},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {57},
YEAR = {2001},
NUMBER = {1},
PAGES = {23--46},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {57M50 (53C22)},
MRNUMBER = {1871490},
MRREVIEWER = {Colin C. Adams},
URL = {http://projecteuclid.org/euclid.jdg/1090348088},
ZBLNUMBER = {1029.57014},
} -
[GaMM2]
D. Gabai, R. G. Meyerhoff, and P. Milley, "Minimum volume cusped hyperbolic three-manifolds," J. Amer. Math. Soc., vol. 22, iss. 4, pp. 1157-1215, 2009.
@article {GaMM2, MRKEY = {2525782},
AUTHOR = {Gabai, David and Meyerhoff, G. Robert and Milley, Peter},
TITLE = {Minimum volume cusped hyperbolic three-manifolds},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {22},
YEAR = {2009},
NUMBER = {4},
PAGES = {1157--1215},
ISSN = {0894-0347},
MRCLASS = {57M50 (51M10 51M25)},
MRNUMBER = {2525782},
MRREVIEWER = {Stephan Tillmann},
DOI = {10.1090/S0894-0347-09-00639-0},
ZBLNUMBER = {1204.57013},
} -
[GMT]
D. Gabai, R. G. Meyerhoff, and N. Thurston, "Homotopy hyperbolic 3-manifolds are hyperbolic," Ann. of Math., vol. 157, iss. 2, pp. 335-431, 2003.
@article {GMT, MRKEY = {1973051},
AUTHOR = {Gabai, David and Meyerhoff, G. Robert and Thurston, Nathaniel},
TITLE = {Homotopy hyperbolic 3-manifolds are hyperbolic},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {157},
YEAR = {2003},
NUMBER = {2},
PAGES = {335--431},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {57M50 (57-04 57N10)},
MRNUMBER = {1973051},
MRREVIEWER = {Darryl McCullough},
DOI = {10.4007/annals.2003.157.335},
ZBLNUMBER = {1052.57019},
} -
[GMineq] F. W. Gehring and G. J. Martin, "Some universal constraints for discrete Möbius groups," in Paul Halmos, New York: Springer-Verlag, 1991, pp. 205-220.
@incollection {GMineq, MRKEY = {1113275},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {Some universal constraints for discrete {M}öbius groups},
BOOKTITLE = {Paul {H}almos},
PAGES = {205--220},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1991},
MRCLASS = {30F35},
MRNUMBER = {1113275},
MRREVIEWER = {Peter J. Nicholls},
ZBLNUMBER = {0784.30038},
} -
[GM1]
F. W. Gehring and G. J. Martin, "On the Margulis constant for Kleinian groups. I," Ann. Acad. Sci. Fenn. Math., vol. 21, iss. 2, pp. 439-462, 1996.
@article {GM1, MRKEY = {1404096},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {On the {M}argulis constant for {K}leinian groups. {I}},
JOURNAL = {Ann. Acad. Sci. Fenn. Math.},
FJOURNAL = {Academiæ Scientiarum Fennicæ . Annales. Mathematica},
VOLUME = {21},
YEAR = {1996},
NUMBER = {2},
PAGES = {439--462},
ISSN = {0066-1953},
MRCLASS = {30F40 (20H10 57N10)},
MRNUMBER = {1404096},
MRREVIEWER = {Feng Luo},
ZBLNUMBER = {0854.30031},
URL = {http://www.acadsci.fi/mathematica/Vol21/gehring.html},
} -
[GM2]
F. W. Gehring and G. J. Martin, "Axial distances in discrete Möbius groups," Proc. Nat. Acad. Sci. U.S.A., vol. 89, iss. 6, pp. 1999-2001, 1992.
@article {GM2, MRKEY = {1154948},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {Axial distances in discrete {M}öbius groups},
JOURNAL = {Proc. Nat. Acad. Sci. U.S.A.},
FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
VOLUME = {89},
YEAR = {1992},
NUMBER = {6},
PAGES = {1999--2001},
ISSN = {0027-8424},
CODEN = {PNASA6},
MRCLASS = {30F40 (20H10 57S30)},
MRNUMBER = {1154948},
DOI = {10.1073/pnas.89.6.1999},
ZBLNUMBER = {0753.30030},
} -
[GM3]
F. W. Gehring and G. J. Martin, "Commutators, collars and the geometry of Möbius groups," J. Anal. Math., vol. 63, pp. 175-219, 1994.
@article {GM3, MRKEY = {1269219},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {Commutators, collars and the geometry of {M}öbius groups},
JOURNAL = {J. Anal. Math.},
FJOURNAL = {Journal d'Analyse Mathématique},
VOLUME = {63},
YEAR = {1994},
PAGES = {175--219},
ISSN = {0021-7670},
CODEN = {JOAMAV},
MRCLASS = {30F40},
MRNUMBER = {1269219},
MRREVIEWER = {Jane Gilman},
DOI = {10.1007/BF03008423},
ZBLNUMBER = {0799.30034},
} -
[GM5] F. W. Gehring and G. J. Martin, "On the minimal volume hyperbolic $3$-orbifold," Math. Res. Lett., vol. 1, iss. 1, pp. 107-114, 1994.
@article {GM5, MRKEY = {1258496},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {On the minimal volume hyperbolic {$3$}-orbifold},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {1},
YEAR = {1994},
NUMBER = {1},
PAGES = {107--114},
ISSN = {1073-2780},
MRCLASS = {30F40 (20H10 57M50)},
MRNUMBER = {1258496},
ZBLNUMBER = {0834.30029},
} -
[GM6]
F. W. Gehring and G. J. Martin, "$6$-torsion and hyperbolic volume," Proc. Amer. Math. Soc., vol. 117, iss. 3, pp. 727-735, 1993.
@article {GM6, MRKEY = {1116260},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {{$6$}-torsion and hyperbolic volume},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {117},
YEAR = {1993},
NUMBER = {3},
PAGES = {727--735},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {30F40 (20H10 57S30)},
MRNUMBER = {1116260},
MRREVIEWER = {Ara Basmajian},
DOI = {10.2307/2159135},
ZBLNUMBER = {0790.30032},
} -
[GM7]
F. W. Gehring and G. J. Martin, "The volume of hyperbolic $3$-folds with $p$-torsion, $p\ge6$," Quart. J. Math. Oxford Ser., vol. 50, iss. 197, pp. 1-12, 1999.
@article {GM7, MRKEY = {1673252},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {The volume of hyperbolic {$3$}-folds with {$p$}-torsion, {$p\ge6$}},
JOURNAL = {Quart. J. Math. Oxford Ser.},
FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
VOLUME = {50},
YEAR = {1999},
NUMBER = {197},
PAGES = {1--12},
ISSN = {0033-5606},
CODEN = {QJMAAT},
MRCLASS = {57M50 (20H15 57S30)},
MRNUMBER = {1673252},
MRREVIEWER = {Leonid Potyagailo},
DOI = {10.1093/qjmath/50.197.1},
ZBLNUMBER = {0926.30027},
} -
[GM8]
F. W. Gehring and G. J. Martin, "Precisely invariant collars and the volume of hyperbolic $3$-folds," J. Differential Geom., vol. 49, iss. 3, pp. 411-435, 1998.
@article {GM8, MRKEY = {1669657},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {Precisely invariant collars and the volume of hyperbolic {$3$}-folds},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {49},
YEAR = {1998},
NUMBER = {3},
PAGES = {411--435},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {57M50 (57N10)},
MRNUMBER = {1669657},
MRREVIEWER = {Colin C. Adams},
URL = {http://projecteuclid.org/euclid.jdg/1214461106},
ZBLNUMBER = {0989.57010},
} -
[GM10] F. W. Gehring and G. J. Martin, "$(p,q,r)$-Kleinian groups and the Margulis constant," in Complex Analysis and Dynamical Systems II, Providence, RI: Amer. Math. Soc., 2005, vol. 382, pp. 149-169.
@incollection {GM10, MRKEY = {2175884},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {{$(p,q,r)$}-{K}leinian groups and the {M}argulis constant},
BOOKTITLE = {Complex Analysis and Dynamical Systems {II}},
SERIES = {Contemp. Math.},
VOLUME = {382},
PAGES = {149--169},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
MRCLASS = {30F40 (20H10)},
MRNUMBER = {2175884},
MRREVIEWER = {Jack O. Button},
ZBLNUMBER = {1088.30044},
} -
[GMSTG]
F. W. Gehring and G. J. Martin, "Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group," Ann. of Math., vol. 170, iss. 1, pp. 123-161, 2009.
@article {GMSTG, MRKEY = {2521113},
AUTHOR = {Gehring, Frederick W. and Martin, G. J.},
TITLE = {Minimal co-volume hyperbolic lattices. {I}. {T}he spherical points of a {K}leinian group},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {170},
YEAR = {2009},
NUMBER = {1},
PAGES = {123--161},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {57M50 (20H10 30F40 57R18)},
MRNUMBER = {2521113},
MRREVIEWER = {Jack O. Button},
DOI = {10.4007/annals.2009.170.123},
ZBLNUMBER = {1171.30014},
} -
[GMMR]
F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, "Arithmeticity, discreteness and volume," Trans. Amer. Math. Soc., vol. 349, iss. 9, pp. 3611-3643, 1997.
@article {GMMR, MRKEY = {1433117},
AUTHOR = {Gehring, Frederick W. and Maclachlan, C. and Martin, G. J. and Reid, A. W.},
TITLE = {Arithmeticity, discreteness and volume},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {349},
YEAR = {1997},
NUMBER = {9},
PAGES = {3611--3643},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {57M50 (11F06 20H15)},
MRNUMBER = {1433117},
MRREVIEWER = {Kerry N. Jones},
DOI = {10.1090/S0002-9947-97-01989-2},
ZBLNUMBER = {0889.30031},
} -
[Hur]
A. Hurwitz, "Ueber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., vol. 41, pp. 403-442, 1892.
@article{Hur,
author={Hurwitz, A.},
TITLE={{U}eber algebraische {G}ebilde mit eindeutigen {T}ransformationen in sich},
JOURNAL={Math. Ann.},
VOLUME={41},
YEAR={1892},
PAGES={403--442},
MRNUMBER = {1510753},
JFMNUMBER = {24.0380.02},
DOI = {10.1007/BF01443420},
} -
[I] T. Inada, "An elementary proof of a theorem of Margulis for Kleinian groups," Math. J. Okayama Univ., vol. 30, pp. 177-186, 1988.
@article {I, MRKEY = {0976740},
AUTHOR = {Inada, Tomio},
TITLE = {An elementary proof of a theorem of {M}argulis for {K}leinian groups},
JOURNAL = {Math. J. Okayama Univ.},
FJOURNAL = {Mathematical Journal of Okayama University},
VOLUME = {30},
YEAR = {1988},
PAGES = {177--186},
ISSN = {0030-1566},
CODEN = {MJOKAP},
MRCLASS = {30F40 (20H10 22E40)},
MRNUMBER = {0976740},
MRREVIEWER = {C. Maclachlan},
ZBLNUMBER = {0674.30036},
} -
[Jorgensen]
T. Jørgensen, "On discrete groups of Möbius transformations," Amer. J. Math., vol. 98, iss. 3, pp. 739-749, 1976.
@article {Jorgensen, MRKEY = {0427627},
AUTHOR = {Jørgensen, Troels},
TITLE = {On discrete groups of {M}öbius transformations},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {98},
YEAR = {1976},
NUMBER = {3},
PAGES = {739--749},
ISSN = {0002-9327},
MRCLASS = {30A58},
MRNUMBER = {0427627},
MRREVIEWER = {V. Chuckrow},
ZBLNUMBER = {0336.30007},
DOI = {10.2307/2373814},
} -
[Jor2] T. Jørgensen, "Commutators in ${ SL}(2, {\bf C})$," in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Princeton, N.J.: Princeton Univ. Press, 1981, vol. 97, pp. 301-303.
@incollection {Jor2, MRKEY = {0624821},
AUTHOR = {Jørgensen, Troels},
TITLE = {Commutators in {${\rm SL}(2, {\bf C})$}},
BOOKTITLE = {Riemann Surfaces and Related Topics: {P}roceedings of the 1978 {S}tony {B}rook {C}onference},
VENUE={{S}tate {U}niv. {N}ew {Y}ork, {S}tony {B}rook, {N}.{Y}., 1978},
SERIES = {Ann. of Math. Stud.},
VOLUME = {97},
PAGES = {301--303},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1981},
MRCLASS = {20G20 (30F99)},
MRNUMBER = {0624821},
ZBLNUMBER = {0459.20044},
} -
[JR]
K. N. Jones and A. W. Reid, "Minimal index torsion-free subgroups of Kleinian groups," Math. Ann., vol. 310, iss. 2, pp. 235-250, 1998.
@article {JR, MRKEY = {1602004},
AUTHOR = {Jones, Kerry N. and Reid, Alan W.},
TITLE = {Minimal index torsion-free subgroups of {K}leinian groups},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {310},
YEAR = {1998},
NUMBER = {2},
PAGES = {235--250},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {57M50 (20H10 30F40 57S30)},
MRNUMBER = {1602004},
MRREVIEWER = {Bruno Zimmermann},
DOI = {10.1007/s002080050147},
ZBLNUMBER = {0890.57016},
} -
[Kirby] "Problems in low-dimensional topology," in Geometric Topology, Providence, RI: Amer. Math. Soc., 1997, vol. 2, pp. 35-473.
@incollection {Kirby, MRKEY = {1470751},
TITLE = {Problems in low-dimensional topology},
BOOKTITLE = {Geometric Topology},
VENUE={{A}thens, {GA},
1993},
SERIES = {AMS/IP Stud. Adv. Math.},
VOLUME = {2},
PAGES = {35--473},
NOTE = {Edited by Rob Kirby},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1997},
MRCLASS = {57-02},
MRNUMBER = {1470751},
} -
[Kn] A. W. Knapp, "Doubly generated Fuchsian groups," Michigan Math. J., vol. 15, pp. 289-304, 1969.
@article {Kn, MRKEY = {0248231},
AUTHOR = {Knapp, A. W.},
TITLE = {Doubly generated {F}uchsian groups},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {15},
YEAR = {1969},
PAGES = {289--304},
ISSN = {0026-2285},
MRCLASS = {20.65},
MRNUMBER = {0248231},
MRREVIEWER = {L. Greenberg},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0167.07002},
} -
[Macbeath] A. M. Macbeath, "On a theorem of Hurwitz," Proc. Glasgow Math. Assoc., vol. 5, pp. 90-96 (1961), 1961.
@article {Macbeath, MRKEY = {0146724},
AUTHOR = {Macbeath, A. M.},
TITLE = {On a theorem of {H}urwitz},
JOURNAL = {Proc. Glasgow Math. Assoc.},
VOLUME = {5},
YEAR = {1961},
PAGES = {90--96 (1961)},
MRCLASS = {14.20 (14.15)},
MRNUMBER = {0146724},
MRREVIEWER = {R. H. Fox},
ZBLNUMBER = {0134.16603},
} -
[Ma] A. Marden, "Universal properties of Fuchsian groups in the Poincaré metric," in Discontinuous Groups and Riemann surfaces, Princeton, N.J.: Princeton Univ. Press, 1974, pp. 315-339. ann. of math. studies, no. 79.
@incollection {Ma, MRKEY = {0379837},
AUTHOR = {Marden, Albert},
TITLE = {Universal properties of {F}uchsian groups in the {P}oincaré metric},
BOOKTITLE = {Discontinuous Groups and {R}iemann surfaces},
VENUE={{P}roc. {C}onf., {U}niv. {M}aryland, {C}ollege {P}ark, {M}d., 1973},
PAGES = {315--339. Ann. of Math. Studies, No. 79},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1974},
MRCLASS = {30A58},
MRNUMBER = {0379837},
MRREVIEWER = {William Harvey},
ZBLNUMBER = {0311.30024},
} -
[Mar] G. A. Margulis, "Discrete groups of motions of manifolds of nonpositive curvature," in Proceedings of the International Congress of Mathematicians, Vol. 2, 1975, pp. 21-34.
@inproceedings {Mar, MRKEY = {0492072},
AUTHOR = {Margulis, G. A.},
TITLE = {Discrete groups of motions of manifolds of nonpositive curvature},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 2},
VENUE={{V}ancouver, {B}.{C}., 1974},
PAGES = {21--34},
PUBLISHER = {Canad. Math. Congress, Montreal, Que.},
YEAR = {1975},
MRCLASS = {22E40},
MRNUMBER = {0492072},
MRREVIEWER = {J. S. Joel},
} -
[pack]
T. H. Marshall and G. J. Martin, "Cylinder and horoball packing in hyperbolic space," Ann. Acad. Sci. Fenn. Math., vol. 30, iss. 1, pp. 3-48, 2005.
@article {pack, MRKEY = {2140297},
AUTHOR = {Marshall, T. H. and Martin, G. J.},
TITLE = {Cylinder and horoball packing in hyperbolic space},
JOURNAL = {Ann. Acad. Sci. Fenn. Math.},
FJOURNAL = {Annales Academiæ Scientiarium Fennicæ . Mathematica},
VOLUME = {30},
YEAR = {2005},
NUMBER = {1},
PAGES = {3--48},
ISSN = {1239-629X},
MRCLASS = {52C17 (30F45 51M09 57M50 57N10)},
MRNUMBER = {2140297},
MRREVIEWER = {Peter A. H{ä}st{ö}},
ZBLNUMBER = {1079.52011},
URL = {http://www.acadsci.fi/mathematica/Vol30/martin.html},
} -
[MM]
T. H. Marshall and G. J. Martin, "Volumes of hyperbolic 3-manifolds. Notes on a paper of D. Gabai, G. Meyerhoff, and P. Milley, “Volumes of tubes in hyperbolic 3-manifolds” [J. Differential Geom. 57 (2001), no. 1, 23–46; MR1871490]," Conform. Geom. Dyn., vol. 7, pp. 34-48, 2003.
@article {MM, MRKEY = {1992036},
AUTHOR = {Marshall, T. H. and Martin, G. J.},
TITLE = {Volumes of hyperbolic 3-manifolds. {N}otes on a paper of {D}. {G}abai, {G}. {M}eyerhoff, and {P}. {M}illey, ``{V}olumes of tubes in hyperbolic 3-manifolds'' [{J}. {D}ifferential {G}eom. {\bf 57} (2001), no. 1, 23--46; {MR}1871490]},
JOURNAL = {Conform. Geom. Dyn.},
FJOURNAL = {Conformal Geometry and Dynamics. An Electronic Journal of the American Mathematical Society},
VOLUME = {7},
YEAR = {2003},
PAGES = {34--48},
ISSN = {1088-4173},
MRCLASS = {57M50 (53C22)},
MRNUMBER = {1992036},
DOI = {10.1090/S1088-4173-03-00081-X},
ZBLNUMBER = {1065.57017},
} -
[Mart] G. J. Martin, "On the geometry of Kleinian groups," in Quasiconformal Mappings and Analysis, New York: Springer-Verlag, 1998, pp. 253-274.
@incollection {Mart, MRKEY = {1488454},
AUTHOR = {Martin, Gaven J.},
TITLE = {On the geometry of {K}leinian groups},
BOOKTITLE = {Quasiconformal Mappings and Analysis},
VENUE={{A}nn {A}rbor, {MI},
1995},
PAGES = {253--274},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1998},
MRCLASS = {30F40},
MRNUMBER = {1488454},
MRREVIEWER = {I. Kra},
ZBLNUMBER = {0886.30030},
} -
[Mart2] G. J. Martin, "The volume of regular tetrahedra and sphere packing in hyperbolic $3$-space," Math. 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},
JOURNAL = {Math. Chronicle},
FJOURNAL = {Mathematical Chronicle},
VOLUME = {20},
YEAR = {1991},
PAGES = {127--147},
ISSN = {0581-1155},
CODEN = {MTHCB3},
MRCLASS = {52C17 (52A38 57S30)},
MRNUMBER = {1137879},
MRREVIEWER = {A. Florian},
ZBLNUMBER = {0761.52012},
} -
[M] B. Maskit, Kleinian groups, New York: Springer-Verlag, 1988.
@book {M, MRKEY = {0959135},
AUTHOR = {Maskit, Bernard},
TITLE = {Kleinian groups},
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 = {0959135},
MRREVIEWER = {William Abikoff},
ZBLNUMBER = {0627.30039},
} -
[Mey]
R. Meyerhoff, "Sphere-packing and volume in hyperbolic $3$-space," Comment. Math. Helv., vol. 61, iss. 2, pp. 271-278, 1986.
@article {Mey, MRKEY = {0856090},
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 = {0856090},
MRREVIEWER = {A. Florian},
DOI = {10.1007/BF02621915},
ZBLNUMBER = {0611.57010},
} -
[Mostow]
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 = {0236383},
MRREVIEWER = {R. J. Cannon},
ZBLNUMBER = {0189.09402},
DOI = {10.1007/BF02684590},
} -
[P]
A. Przeworski, "A universal upper bound on density of tube packings in hyperbolic space," J. Differential Geom., vol. 72, iss. 1, pp. 113-127, 2006.
@article {P, MRKEY = {2215457},
AUTHOR = {Przeworski, Andrew},
TITLE = {A universal upper bound on density of tube packings in hyperbolic space},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {72},
YEAR = {2006},
NUMBER = {1},
PAGES = {113--127},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {57M50 (57N10)},
MRNUMBER = {2215457},
MRREVIEWER = {Irene Scorza},
URL = {http://projecteuclid.org/euclid.jdg/1143593127},
ZBLNUMBER = {1098.52005},
} -
[Rat] J. G. Ratcliffe, Foundations of hyperbolic manifolds, New York: Springer-Verlag, 1994.
@book {Rat, MRKEY = {1299730},
AUTHOR = {Ratcliffe, John G.},
TITLE = {Foundations of hyperbolic manifolds},
SERIES = {Graduate Texts in Mathematics},
NUMBER = {149},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1994},
PAGES = {xii+747},
ISBN = {0-387-94348-X},
MRCLASS = {57M50 (20H10 30F40 51M10)},
MRNUMBER = {1299730},
MRREVIEWER = {Colin C. Adams},
ZBLNUMBER = {0809.51001},
} -
[Rolfsen] D. Rolfsen, Knots and Links, Berkeley, CA: Publish or Perish, 1976, vol. 7.
@book {Rolfsen, MRKEY = {0515288},
AUTHOR = {Rolfsen, Dale},
TITLE = {Knots and Links},
SERIES = {Math. Lecture Series},
VOLUME={7},
PUBLISHER = {Publish or Perish},
ADDRESS = {Berkeley, CA},
YEAR = {1976},
PAGES = {ix+439},
MRCLASS = {55-01},
MRNUMBER = {0515288},
ZBLNUMBER = {0339.55004},
} -
[Selberg] A. Selberg, "On discontinuous groups in higher-dimensional symmetric spaces," in Contributions to Function Theory, Bombay: Tata Institute of Fundamental Research, 1960, pp. 147-164.
@incollection {Selberg, MRKEY = {0130324},
AUTHOR = {Selberg, Atle},
TITLE = {On discontinuous groups in higher-dimensional symmetric spaces},
BOOKTITLE = {Contributions to Function Theory},
VENUE={Internat. {C}olloq. {F}unction {T}heory, {B}ombay, 1960},
PAGES = {147--164},
PUBLISHER = {Tata Institute of Fundamental Research},
ADDRESS = {Bombay},
YEAR = {1960},
MRCLASS = {20.65 (22.70)},
MRNUMBER = {0130324},
MRREVIEWER = {E. Grosswald},
ZBLNUMBER = {0201.36603},
} -
@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 = {0009959},
MRREVIEWER = {N. Jacobson},
ZBLNUMBER = {0061.04504},
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 = {0014088},
MRREVIEWER = {S. A. Jennings},
ZBLNUMBER = {0061.04505},
} -
[V] È. B. Vinberg, "Hyperbolic groups of reflections," Uspekhi Mat. Nauk, vol. 40, iss. 1(241), pp. 29-66, 255, 1985.
@article {V, MRKEY = {0783604},
AUTHOR = {Vinberg, {È}. B.},
TITLE = {Hyperbolic groups of reflections},
JOURNAL = {Uspekhi Mat. Nauk},
FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
VOLUME = {40},
YEAR = {1985},
NUMBER = {1(241)},
PAGES = {29--66, 255},
ISSN = {0042-1316},
MRCLASS = {53C20 (20H15 22E40 32M10)},
MRNUMBER = {0783604},
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 = {0630246},
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 = {0630246},
MRREVIEWER = {L. Keen},
URL = {http://projecteuclid.org/euclid.kmj/1138036373},
ZBLNUMBER = {0469.30038},
}