Marshall, Timothy H.; Martin, Gaven J. Minimal co-volume hyperbolic lattices. II: Simple torsion in a Kleinian group. (English) Zbl 1252.30030 Ann. Math. (2) 176, No. 1, 261-301 (2012). 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)] Reviewer: Gerhard Rosenberger (Hamburg) Cited in 2 ReviewsCited in 19 Documents 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) Keywords:hyperbolic lattice; hyperbolic orbifold; Kleinian group; minimal volume; hyperbolic 3-space; arithmetic lattice Citations:Zbl 0061.04504; Zbl 1171.30014 × Cite Format Result Cite Review PDF Full Text: DOI 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. 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