Changing topological type of compression bodies through cone manifolds

Abstract: Generically, small deformations of cone manifold holonomy groups have wildly uncontrolled global geometry. We give a short concrete example showing that it is possible to deform complete hyperbolic metrics on a thickened genus $2$ surface to complete hyperbolic metrics on the genus two handlebody with a single unknotted cusp drilled out via cone manifolds of prescribed singular structure. In other words, there exists a method to construct smooth curves in the character variety of $ \pi_1(S_{2,0}) $ which join open sets parameterising discrete groups (quasi-conformal deformation spaces) through indiscrete groups where the indiscreteness arises in a very controlled, local, way: a cone angle increasing along a fixed singular locus.