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The boundary of the deformation space of the fundamental group of some hyperbolic 3–manifolds fibering over the circle

Leonid Potyagailo

Geometry & Topology Monographs 1 (1998) 479–492
DOI: 10.2140/gtm.1998.1.479

arXiv: math.GT/9811181

Abstract
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By using Thurston’s bending construction we obtain a sequence of faithful discrete representations ρn of the fundamental group of a closed hyperbolic 3–manifold fibering over the circle into the isometry group Iso 4 of the hyperbolic space 4. The algebraic limit of ρn contains a finitely generated subgroup F whose 3–dimensional quotient Ω(F)F has infinitely generated fundamental group, where Ω(F) is the discontinuity domain of F acting on the sphere at infinity S3 = 4. Moreover F is isomorphic to the fundamental group of a closed surface and contains infinitely many conjugacy classes of maximal parabolic subgroups.

Keywords
discrete (Kleinian) subgroups, deformation spaces, hyperbolic 4–manifolds, conformally flat 3–manifolds, surface bundles over the circle
Mathematical Subject Classification
Primary: 57M10, 30F40, 20H10
Secondary: 57S30, 57M05, 30F10, 30F35
References
Publication
Received: 20 November 1997
Revised: 7 November 1998
Published: 17 November 1998
Authors
Leonid Potyagailo
Département de Mathématiques
Université de Lille 1
59655 Villeneuve d’Ascq
France