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Cusps, triangle groups and hyperbolic 3-folds

Part of: Topological manifolds Riemann surfaces Other groups of matrices

Published online by Cambridge University Press:  09 April 2009

Marston D. E. Conder  and
Gaven J. Martin
Marston D. E. Conder
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019 Auckland, NEW ZEALAND, email: conder@mat.aukuni.ac.nz
Gaven J. Martin
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019 Auckland, NEW ZEALAND, email: martin@mat.aukuni.ac.nz
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Abstract

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We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

MSC classification

Secondary: 20H10: Fuchsian groups and their generalizations 30F40: Kleinian groups 57N10: Topology of general $3$-manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 2 , October 1993 , pp. 149 - 182
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Adams, C. C., ‘The noncompact hyperbolic 3-manifold of minimal volume’, Proc. Amer. Math. Soc. 100 (1987), 601–606.CrossRefGoogle Scholar
[2]Adams, C. C., ‘Limit volumes of hyperbolic three-orbifolds’, J. Differential Geom. 34 (1991), 115–142.CrossRefGoogle Scholar
[3]Beardon, A., The geometry of discrete groups (Springer-Verlag, 1983).CrossRefGoogle Scholar
[4]Brooks, R., and Matelski, P., ‘Collars in Kleinian groups’, Duke Math. J. 49 (1982), 163–182.CrossRefGoogle Scholar
[5]Chinburg, T. and Friedman, E., ‘The smallest arithmetic hyperbolic three-orbifold’, Invent. Math. 86 (1986), 507–527.CrossRefGoogle Scholar
[6]Coxeter, H. S. M., ‘The functions of Schläfli and Lobachevshy’, Quart. J. Math. Ser. (2) 6 (1935), 13–29.CrossRefGoogle Scholar
[7]Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (Springer-Verlag, Berlin, 1979).Google Scholar
[8]Gehring, F. W. and Martin, G. J., ‘6-Torsion and hyperbolic volume’, Proc. Amer. Math. Soc., to appear.Google Scholar
[9]Gehring, F. W., ‘Commutators, collars and the geometry of Möbius groups’, preprint.Google Scholar
[10]Gehring, F. W., ‘Torsion and volume in hyperbolic 3-folds’, preprint.Google Scholar
[11]Gehring, F. W., ‘Inequalities for matrices and Möbius transformations’, J. Reine Angew. Math. 418 (1991), 31–76.Google Scholar
[12]Hodgson, C., Degeneration and regeneration of hyperbolic structures on three-manifolds (Ph. D. Thesis, Princeton University, 1986).Google Scholar
[13]Jørgensen, T., ‘On discrete groups of Möbius transformations’, Amer. J. Math. 98 (1976), 739–749.CrossRefGoogle Scholar
[14]Kojima, S. and Miyamoto, Y., ‘The smallest hyperbolic 3-manifolds with totally geodesic boundary’, J. Differential Geom. 34 (1991), 175–192.CrossRefGoogle Scholar
[15]Lanner, F., ‘On complexes with transitive groups of automorphisms’, Comm. Sém. Math. Univ. Lund. 11 (1950), 41.Google Scholar
[16]Martin, G. J., ‘The volume of regular tetrahedra and sphere packing in hyperbolic space’, Math. Chronicle 20 (1991), 127–147.Google Scholar
[17]Maskit, B., Kleinian groups (Springer-Verlag, Berlin, 1987).CrossRefGoogle Scholar
[18]Meyerhoff, R., ‘The cusped hyperbolic 3-orbifold of minimum volume’, Bull. Amer. Math. Soc. 13 (1985), 154–156.CrossRefGoogle Scholar
[19]Meyerhoff, R., ‘A lower bound for the volume of hyperbolic 3-orbifolds’, Duke Math. J. 57 (1988), 185–203.CrossRefGoogle Scholar
[20]Mostow, G. D., ‘Quasiconformal mappings in n-space and the rigidity of hyperbolic space forms’, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104.CrossRefGoogle Scholar
[21]Thurston, W. P., The geometry and topology of three-manifolds (Lecture notes, Princeton University, 1977).Google Scholar
[22]Vinberg, E. B., ‘Discrete groups generated by reflections in Lobacevski spaces’, Math. Sb., 72 (114) (1967), 429–444.CrossRefGoogle Scholar