# zbMATH — the first resource for mathematics

Minimal co-volume hyperbolic lattices. II: Simple torsion in a Kleinian group. (English) Zbl 1252.30030
Authors’ abstract: “This paper represents the final step in solving the problem, posed by C. L. Siegel in [Ann. Math. (2) 44, 674–689 (1943; Zbl 0061.04504)], 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.”
For Part I see [F. W. Gehring and the second author, Ann. Math. (2) 170, No. 1, 123–161 (2009; Zbl 1171.30014)]

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57M50 General geometric structures on low-dimensional manifolds 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
##### Citations:
Zbl 0061.04504; Zbl 1171.30014
Full Text:
##### References:
 [1] 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. · Zbl 0365.30011 [2] E. M. Andreev, ”Convex polyhedra in Loba,” Mat. Sb., vol. 81 (123), pp. 445-478, 1970. [3] A. F. Beardon, The Geometry of Discrete Groups, New York: Springer-Verlag, 1983, vol. 91. · Zbl 0528.30001 [4] 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. · Zbl 0473.57003 [5] K. Böröczky, ”Packing of spheres in spaces of constant curvature,” Acta Math. Acad. Sci. Hungar., vol. 32, iss. 3-4, pp. 243-261, 1978. · Zbl 0422.52011 [6] C. Cao, ”On three generator Möbius groups,” New Zealand J. Math., vol. 23, iss. 2, pp. 111-120, 1994. · Zbl 0839.30048 [7] C. Cao, ”Some trace inequalities for discrete groups of Möbius transformations,” Proc. Amer. Math. Soc., vol. 123, iss. 12, pp. 3807-3815, 1995. · Zbl 0843.30037 [8] 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. · Zbl 0884.30034 [9] L. Carleson and T. W. Gamelin, Complex Dynamics, New York: Springer-Verlag, 1993. · Zbl 0782.30022 [10] T. Chinburg and E. Friedman, ”The smallest arithmetic hyperbolic three-orbifold,” Invent. Math., vol. 86, iss. 3, pp. 507-527, 1986. · Zbl 0643.57011 [11] 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. · Zbl 1104.20035 [12] 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. · Zbl 0769.57010 [13] 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. · Zbl 0713.30044 [14] 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. · Zbl 1029.57014 [15] 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. · Zbl 1204.57013 [16] 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. · Zbl 1052.57019 [17] 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. · Zbl 0784.30038 [18] 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. · Zbl 0854.30031 [19] 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. · Zbl 0753.30030 [20] F. W. Gehring and G. J. Martin, ”Commutators, collars and the geometry of Möbius groups,” J. Anal. Math., vol. 63, pp. 175-219, 1994. · Zbl 0799.30034 [21] 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. · Zbl 0834.30029 [22] F. W. Gehring and G. J. Martin, ”$$6$$-torsion and hyperbolic volume,” Proc. Amer. Math. Soc., vol. 117, iss. 3, pp. 727-735, 1993. · Zbl 0790.30032 [23] F. W. Gehring and G. J. Martin, ”The volume of hyperbolic $$3$$-folds with $$p$$-torsion, $$p\geq6$$,” Quart. J. Math. Oxford Ser., vol. 50, iss. 197, pp. 1-12, 1999. · Zbl 0926.30027 [24] 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. · Zbl 0989.57010 [25] 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. · Zbl 1088.30044 [26] 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. · Zbl 1171.30014 [27] 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. · Zbl 0889.30031 [28] A. Hurwitz, ”Ueber algebraische Gebilde mit eindeutigen Transformationen in sich,” Math. Ann., vol. 41, pp. 403-442, 1892. · JFM 24.0380.02 [29] T. Inada, ”An elementary proof of a theorem of Margulis for Kleinian groups,” Math. J. Okayama Univ., vol. 30, pp. 177-186, 1988. · Zbl 0674.30036 [30] T. Jørgensen, ”On discrete groups of Möbius transformations,” Amer. J. Math., vol. 98, iss. 3, pp. 739-749, 1976. · Zbl 0336.30007 [31] T. Jørgensen, ”Commutators in $${ SL}(2, {\mathbf C})$$,” 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. · Zbl 0459.20044 [32] 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. · Zbl 0890.57016 [33] ”Problems in low-dimensional topology,” in Geometric Topology, Providence, RI: Amer. Math. Soc., 1997, vol. 2, pp. 35-473. · Zbl 0888.57014 [34] A. W. Knapp, ”Doubly generated Fuchsian groups,” Michigan Math. J., vol. 15, pp. 289-304, 1969. · Zbl 0167.07002 [35] A. M. Macbeath, ”On a theorem of Hurwitz,” Proc. Glasgow Math. Assoc., vol. 5, pp. 90-96 (1961), 1961. · Zbl 0134.16603 [36] 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. · Zbl 0311.30024 [37] 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. [38] 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. · Zbl 1079.52011 [39] 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. · Zbl 1065.57017 [40] G. J. Martin, ”On the geometry of Kleinian groups,” in Quasiconformal Mappings and Analysis, New York: Springer-Verlag, 1998, pp. 253-274. · Zbl 0886.30030 [41] G. J. Martin, ”The volume of regular tetrahedra and sphere packing in hyperbolic $$3$$-space,” Math. Chronicle, vol. 20, pp. 127-147, 1991. · Zbl 0761.52012 [42] B. Maskit, Kleinian groups, New York: Springer-Verlag, 1988. · Zbl 0627.30039 [43] R. Meyerhoff, ”Sphere-packing and volume in hyperbolic $$3$$-space,” Comment. Math. Helv., vol. 61, iss. 2, pp. 271-278, 1986. · Zbl 0611.57010 [44] 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. · Zbl 0189.09402 [45] 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. · Zbl 1098.52005 [46] J. G. Ratcliffe, Foundations of hyperbolic manifolds, New York: Springer-Verlag, 1994. · Zbl 0809.51001 [47] D. Rolfsen, Knots and Links, Berkeley, CA: Publish or Perish, 1976, vol. 7. · Zbl 0339.55004 [48] 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. · Zbl 0201.36603 [49] C. L. Siegel, ”Discontinuous groups,” Ann. of Math., vol. 44, pp. 674-689, 1943. · Zbl 0061.04504 [50] C. L. Siegel, ”Some remarks on discontinuous groups,” Ann. of Math., vol. 46, pp. 708-718, 1945. · Zbl 0061.04505 [51] È. B. Vinberg, ”Hyperbolic groups of reflections,” Uspekhi Mat. Nauk, vol. 40, iss. 1(241), pp. 29-66, 255, 1985. [52] A. Yamada, ”On Marden’s universal constant of Fuchsian groups,” Kodai Math. J., vol. 4, iss. 2, pp. 266-277, 1981. · Zbl 0469.30038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.