Geometry & Topology Monographs 1 (1998), The Epstein Birthday Schrift, paper no. 23, pages 479-492.

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

Leonid Potyagailo

Abstract. By using Thurston's bending construction we obtain a sequence of faithful discrete representations \rho _n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of \rho _n contains a finitely generated subgroup F whose 3-dimensional quotient \Omega (F)/F has infinitely generated fundamental group, where \Omega (F) is the discontinuity domain of F acting on the sphere at infinity. 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

AMS subject classification. Primary: 57M10, 30F40, 20H10. Secondary: 57S30, 57M05, 30F10, 30F35.

E-print: arXiv:math.GT/9811181

Submitted: 20 November 1997. (Revised: 7 November 1998.) Published: 17 November 1998.

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Leonid Potyagailo
Departement de Mathematiques
Universite de Lille 1
59655 Villeneuve d'Ascq, France

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