Regenerating hyperbolic cone structures from Nil

Abstract

Let $\mathcal{O}$ be a three-dimensional $Nil$–orbifold, with branching locus a knot $\Sigma$ transverse to the Seifert fibration. We prove that $\mathcal{O}$ is the limit of hyperbolic cone manifolds with cone angle in $\left(\pi -\epsilon ,\pi \right)$. We also study the space of Dehn filling parameters of $\mathcal{O}-\Sigma$. Surprisingly it is not diffeomorphic to the deformation space constructed from the variety of representations of $\mathcal{O}-\Sigma$. As a corollary of this, we find examples of spherical cone manifolds with singular set a knot that are not locally rigid. Those examples have large cone angles.