arXiv: math.GT/9811181

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 $\Omega \left(F\right)∕F$ has infinitely generated fundamental group, where $\Omega \left(F\right)$ is the discontinuity domain of $F$ acting on the sphere at infinity $S_{\infty }^3=\partial {ℍ}^4$. Moreover $F$ is isomorphic to the fundamental group of a closed surface and contains infinitely many conjugacy classes of maximal parabolic subgroups.

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