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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

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 H4 of the hyperbolic space H4 . 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=∂H4. 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: 20H10, 30F40, 57M10

Secondary: 30F10, 30F35, 57M05, 57S30

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