The asymptotic directions of pleating rays in the Maskit embedding

This article was born as an application of the Top Terms’ Relationship proved by the author and Series in [15] and also as generalisation of the analysis made by Series in [26] where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use this formula to determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface Σ 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 Σ is the space of geometrically finite groups on the boundary of quasifuchsian space for which the ‘top’ end is homeomorphic to Σ, while the ‘bottom’ end consists of triply punctured spheres, the remains of Σ when the pants curves have been pinched. Given a projective measured lamination [η] on Σ, the pleating ray P = P[η] is the set of groups in M for which the bending measure pl(G) of the top component ∂C+ of the boundary of the convex core of the associated 3-manifold H3/G is in the class [η].