Convergent Sequences of Hyperbolic 3-Manifolds with Unfaithful Markings.

Let $\Gamma $ be a finitely generated group. To every representation $\rho
 : \Gamma \to Isom ({\bf H}^3) $ with discrete and torsion-free image there
corresponds a hyperbolic $3$-manifold $M_\rho = {\bf H}^3 / \rho (\Gamma) $. I
will present some new results linking the pointwise convergence of a
sequence of such representations with Gromov-Hausdorff convergence of the
corresponding quotient manifolds. A detailed analysis already exists for
sequences of faithful representations; I will give examples that illustrate
the failure of these theorems in the unfaithful setting, and offer some
useful replacements. Joint work with Juan Souto.