Electronic Resource
Springer
Transformation groups
3 (1998), S. 355-374
ISSN:
1531-586X
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We study discrete (Kleinian) subgroups of the isometry group Iso+H 4 of the real hyperbolic space of dimension 4. Suppose that a finitely generated geometrically finite Kleinian groupG⊃Iso+H 4 act discontinuously on a connected set Ω G ⊂S 3 and contains a nontrivial normal finitely generated subgroupF 0◃G of infinite index. We prove that the fundamental group π1(Ω G /F 0) is finitely generated iff Ω G is simply connected. In particular, if there is one such a groupF 0 for which the group π1(Ω G /F 0) is finitely generated, then the same is true for any other nontrivial normal finitely generated subgroupF ofG of infinite index. On the other hand, a number of examples exists of finitely generated Kleinian groups Γ⊂Iso+H 4 for which the fundamental group π1(ΩГ/Г) is not finitely generated if one of our conditions is not satisfied. Using our method, we provide a simplified proof of a recent result of M. Boileau and S. Wang giving an infinite tower of finite coverings of hyperbolic 3-manifolds not fibering over the circle.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01234533
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