The asymptotic directions of pleating rays in the Maskit embedding

TL;DR

This paper analyzes the asymptotic behavior of pleating rays in the Maskit embedding of hyperbolic surfaces, extending previous work to higher-dimensional deformation spaces using the Top Terms' Relationship.

Contribution

It generalizes Series' analysis to higher-dimensional deformation spaces by determining the asymptotic directions of pleating rays in the Maskit embedding.

Findings

Identifies asymptotic directions of pleating rays as bending measures tend to zero.

Utilizes the Top Terms' Relationship to analyze deformation spaces.

Extends prior analysis from one-dimensional to higher-dimensional cases.

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.