Sequences of Hyperbolic $3$-Manifolds with Unfaithful Markings

Sequences of Hyperbolic $3$-Manifolds with Unfaithful Markings

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Ian Biringer, University of Chicago
Fine Hall 314

Let $\Gamma$ be a finitely generated group. To every representation $\rho : \Gamma\to Isom (BH3)$ with discrete and torsion-free image there corresponds a hyperbolic $3$-manifold $M_\rho = BH3 / \rho (\Gamma)$. 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.