Regenerating hyperbolic cone structures from Nil

Let O be a three-dimensional Nil-orbifold, with branching locus a knot Σ transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (π - ε, π). We also study the space of Dehn filling parameters of O - Σ. Surprisingly it is not diffeomorphic to the deformation space constructed from the variety of representations of O - Σ. 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.