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LIMIT SETS AND COMMENSURABILITY OF KLEINIAN GROUPS

Part of: Riemann surfaces

Published online by Cambridge University Press:  02 June 2010

WEN-YUAN YANG  and
YUE-PING JIANG
WEN-YUAN YANG*
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China U.F.R. de Mathématiques, Université de Lille 1, 59655 Villeneuve D’Ascq Cedex, France (email: wyang@math.univ-lille1.fr)
YUE-PING JIANG
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China (email: ypjiang731@163.com)
*
✉For correspondence; e-mail: Wen-Yuang.Yang@math.univ-lille1.fr
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Abstract

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In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups G1 and G2 of an infinite co-volume Kleinian group G⊂Isom(H3) having Λ(G1)=Λ(G2) are commensurable. In particular, we prove that any finitely generated subgroup H of a Kleinian group G⊂Isom(H3) with Λ(H)=Λ(G) is of finite index if and only if H is not a virtually fibered subgroup.

Keywords

commensurabilitygeometrically finitelimit sethyperbolic element

MSC classification

Secondary: 30F40: Kleinian groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 1 - 9
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author is supported by the China-funded Postgraduates Studying Aboard Program for Building Top University. The second author is supported by National Natural Science Foundation of China (No. 10671059) and Doctorate Foundation of the Ministry of Education of China (No. 20060532023).

References

[1]Agol, I., ‘Tameness of hyperbolic 3-manifolds’, Preprint, 2004.Google Scholar
[2]Anderson, J., ‘On the finitely generated intersection property for Kleinian groups’, Complex Var. Theory Appl. 17 (1991), 111–112.Google Scholar
[3]Anderson, J., ‘Intersections of topologically tame subgroups of Kleinian groups’, J. Anal. Math. 65 (1995), 77–94.CrossRefGoogle Scholar
[4]Anderson, J., ‘The limit set intersection theorem for finitely generated Kleinian groups’, Math. Res. Lett. 3 (1996), 675–692.CrossRefGoogle Scholar
[5]Bestvina, M., ‘Questions in geometric group theory’, http://www.math.utah.edu/∼bestvina.Google Scholar
[6]Calegari, D. and Gabai, D., ‘Shrinkwrapping and the taming of hyperbolic 3-manifolds’, J. Amer. Math. Soc. 19 (2006), 385–446.CrossRefGoogle Scholar
[7]Canary, R., ‘A covering theorem for hyperbolic 3-manifolds and its applications’, Topology 35 (1996), 751–778.CrossRefGoogle Scholar
[8]Greenberg, L., ‘Commensurable groups of Moebius transformations’, in: Discontinuous Groups and Riemann Surfaces, Annals of Mathematics Studies, 79 (ed. Greenberg, L.) (Princeton University Press, Princeton, NJ, 1974), pp. 227–237.CrossRefGoogle Scholar
[9]Long, D. and Reid, A., ‘On Fuchsian groups with the same set of axes’, Bull. Lond. Math. Soc. 30 (1998), 533–538.CrossRefGoogle Scholar
[10]McCullough, D., ‘Compact submanifolds of 3-manifolds with boundary’, Q. J. Math. Oxford 37 (1986), 299–307.CrossRefGoogle Scholar
[11]Mess, G., ‘Fuchsian groups with the same simple axes’, Preprint, 1990.Google Scholar
[12]Morgan, J., ‘On Thurston’s uniformization theorem for 3-dimensional manifolds’, in: The Smith Conjecture (eds. Bass, H. and Morgan, J.) (Academic Press, New York, 1984), pp. 37–125.Google Scholar
[13]Susskind, P., ‘An infinitely generated intersection of the hyperbolic group’, Proc. Amer. Math. Soc. 129 (2001), 2643–2646.CrossRefGoogle Scholar
[14]Susskind, P. and Swarup, G., ‘Limit sets of geometrically finite hyperbolic groups’, Amer. J. Math. 114 (1992), 233–250.CrossRefGoogle Scholar
[15]Thurston, W., The Geometry and Topology of 3-manifolds, Lecture Notes (Princeton Mathematics Department, Princeton, NJ, 1978).Google Scholar