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ON GEHRING–MARTIN–TAN GROUPS WITH AN ELLIPTIC GENERATOR

Part of: Other groups of matrices Low-dimensional topology

Published online by Cambridge University Press:  12 May 2016

DUŠAN REPOVŠ  and
ANDREI VESNIN
DUŠAN REPOVŠ*
Affiliation:
Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia email dusan.repovs@guest.arnes.si
ANDREI VESNIN
Affiliation:
Laboratory of Quantum Topology, Chelyabinsk State University, Chalyabinsk, Russia Sobolev Institute of Mathematics, Novosibirsk 630090, Russia email vesnin@math.nsc.ru
*
✉dusan.repovs@guest.arnes.si
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Abstract

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The Gehring–Martin–Tan inequality for two-generator subgroups of $\text{PSL}(2,\mathbb{C})$ is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring–Martin–Tan group if the equality holds for the group $G$. We give a method for constructing Gehring–Martin–Tan groups with a generator of order four and present some examples. These groups arise as groups of finite-volume hyperbolic 3-orbifolds.

Keywords

hyperbolic orbifolddiscreteness conditiontwo-generated group

MSC classification

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 20H15: Other geometric groups, including crystallographic groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 326 - 336
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Atkinson, C. K. and Futer, D., ‘The lowest volume 3-orbifolds with high torsion’, Trans. Amer. Math. Soc. (to appear) doi:10.1090/tran/6920.Google Scholar
Beardon, A. F., The Geometry of Discrete Groups (Springer, New York, 1983).Google Scholar
Boileau, M., Maillot, S. and Porti, J., Three-Dimensional Orbifolds and their Geometric Structures, Panoramas et Synthéses, 15 (Société Mathématique de France, Paris, 2003).Google Scholar
Callahan, J., ‘Jørgensen number and arithmeticity’, Conform. Geom. Dyn. 13 (2009), 160–186.CrossRefGoogle Scholar
Cao, W. and Tan, H., ‘Jørgensen inequality for quaternionic hyperbolic space with elliptic elements’, Bull. Aust. Math. Soc. 81 (2010), 121–131.Google Scholar
Dehn, M., ‘Dei beiden Kleeblattschlingen’, Math. Ann. 75 (1914), 402–413.CrossRefGoogle Scholar
Gehring, F. W. and Martin, G. J., ‘Stability and extremality in Jørgensen’s inequality’, Complex Var. Theory Appl. 12 (1989), 277–282.Google Scholar
Gongopadhyay, K. and Mukherjee, A., ‘Extremality of quaternionic Jørgensen inequality’.arXiv:math1503.08802.Google Scholar
Jørgensen, T., ‘A note on subgroups of SL(2, ℂ)’, Q. J. Math. Oxford Ser. (2) 28(110) (1977), 209–211.Google Scholar
Klimenko, E. and Kopteva, N., ‘Two-generated Kleinian orbifolds’. arXiv:math/0606066.Google Scholar
Magnus, W., ‘Untersuchungen fiber einige unendliche diskontinuierliche Gruppen’, Math. Ann. 105 (1931), 52–74.Google Scholar
Marimoto, K., Sakuma, M. and Yokota, Y., ‘Identifying tunnel number one knots’, J. Math. Soc. Japan 48(4) (1996), 667–688.Google Scholar
Matveev, S., Petronio, C. and Vesnin, A., ‘Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds’, J. Aust. Math. Soc. 86 (2009), 205–219.CrossRefGoogle Scholar
Mednykh, A. and Rasskazov, A., ‘On the structure of the canonical fundamental set for the 2-bridge link orbifolds’, Universität Bielefeld, Preprint, 1998.Google Scholar
Parker, J. R., ‘Shimizu’s lemma for complex hyperbolic space’, Internat. J. Math. 3 (1992), 291–308.Google Scholar
Riley, R., ‘A quadratic parabolic group’, Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288.Google Scholar
Sato, H., ‘The Jørgensen number of the Whitehead link group’, Bol. Soc. Mat. Mexicana (3) 10 (2004), 495–502.Google Scholar
Tan, D., ‘On two-generator discrete groups of Mobius transformations’, Proc. Amer. Math. Soc. 106 (1989), 763–770.Google Scholar
Vesnin, A. and Masley, A., ‘On Jørgensen numbers and their analogues for groups of figure-eight orbifolds’, Sib. Math. J. 55 (2014), 807–816.Google Scholar
Vesnin, A. and Masley, A., ‘Two-generated subgroups of $\text{PSL}(2,\mathbb{C})$ which are extreme for Jørgensen inequality and its analogues’, Proc. Semin. Vector and Tensor Analysis with their Applications to Geometry, Mechanics and Physics, 30 (Moscow State University, 2015), 1–54.Google Scholar
Vesnin, A. and Mednykh, A., ‘Hyperbolic volumes of the Fibonacci manifolds’, Sib. Math. J. 36 (1995), 235–245.CrossRefGoogle Scholar
Vesnin, A. and Mednykh, A., ‘Fibonacci manifolds as two-fold coverings over the three-dimensional sphere and the Meyerhoff–Neumann conjecture’, Sib. Math. J. 37 (1996), 461–467.Google Scholar
Vesnin, A. and Rasskazov, A., ‘Isometries of hyperbolic Fibonacci manifolds’, Sib. Math. J. 40 (1999), 9–22.Google Scholar