The asymptotic directions of pleating rays in the Maskit embedding

Abstract

This article was born as a generalisation of the analysis made by Series, where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use the Top Terms' Relationship proved by the author and Series to determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface S as the bending measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit embedding M of a surface S is the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to S, while the `bottom' end consists of triply punctured spheres, the remains of S when the pants curves have been pinched. Given a projective measured lamination l on S, the pleating ray P is the set of groups in M for which the bending measure of the top component of the boundary of the convex core of the associated 3-manifold is in the projective class of l.