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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
$Iso{\mathbb{H}}^{4}$ of the hyperbolic space
${\mathbb{H}}^{4}$. The algebraic limit
of
${\rho}_{n}$ contains a finitely
generated subgroup
$F$
whose
$3$–dimensional
quotient
$\Omega \left(F\right)\u2215F$
has infinitely generated fundamental group, where
$\Omega \left(F\right)$ is the discontinuity
domain of
$F$ acting on
the sphere at infinity
${S}_{\infty}^{3}=\partial {\mathbb{H}}^{4}$.
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