| Abstrakt: |
We study discrete (Kleinian) subgroups of the isometry group Iso+ H4 of the real hyperbolic space of dimension 4. Suppose that a finitely generated geometrically finite Kleinian group G⊃Iso+ H4 act discontinuously on a connected set Ω G ⊂ S3 and contains a nontrivial normal finitely generated subgroup F0◃G of infinite index. We prove that the fundamental group π1(Ω G/ F0) is finitely generated iff Ω G is simply connected. In particular, if there is one such a group F0 for which the group π1(Ω G/ F0) is finitely generated, then the same is true for any other nontrivial normal finitely generated subgroup F of G of infinite index. On the other hand, a number of examples exists of finitely generated Kleinian groups Γ⊂Iso+ H4 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. [ABSTRACT FROM AUTHOR] |
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