Abstract

We discuss the process of algebraically finding cusps on the boundaries of deformation spaces of kleinian groups. The geometric starting point is an arrangement of circles with prescribed tangencies and relationships under a set of Möbius transformations. These lead to polynomial equations in several complex variables, which may then be numerically solved for the values which describe the cusp. We will go through this process for several deformation spaces corresponding to “plumbing” constructions of Maskit and Kra, and we will present some of the numerical output. The same techniques can also be used to calculate the coherent spiral hexagonal circle packings discovered by Peter Doyle, and we will compare the similarity factors of those packings to cusps on boundaries of deformation spaces.

Introduction

In this paper, we study the theoretical and numerical calculation of maximal cusp groups on the boundary of deformation spaces of kleinian groups. Specifically, we are interested in maximally parabolic groups which allow no deformations with a greater number of classes of parabolic elements.

The foundational study of cusp groups occurred in Bers' paper [Ber70] on boundaries of Teichmüller spaces. There he proved that the cusp groups form a set of measure zero in the boundary, and that there exist boundary groups he termed totally degenerate, for which the ordinary set is a single invariant domain. Then came a windstorm of ideas from Thurston, first in his celebrated Geometry and Topology of 3-Manifolds and then in many subsequent talks and notes [Thu80, Thu82, Thu89, KT90].