Hyperbolic Dehn filling and the shape of the Maskit slice

The Maskit slice is an embedding of the Teichm"uller space of the punctured torus into $\mathbb{C}$. The Maskit slice can also be identified with a deformation space of hyperbolic 3-manifolds; boundaries of these deformation spaces have proven to be a fruitful area of research in hyperbolic geometry. Minsky showed that the boundary of the Maskit slice is a Jordan curve, and Miyachi proved that the boundary is not a quasicircle. We reprove Miyachi’s result using the hyperbolic Dehn filling theorem, and show how this technique could be used to study the shape of other deformation spaces of hyperbolic 3-manifolds.