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

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.