Geometry for Kleinian Groups Generated by a Parabolic Pair

TL;DR

This paper introduces a computational framework using Farey recursive polynomials and Sakuma-Weeks triangulations to analyze the hyperbolic geometry of 2-bridge link complements and Kleinian groups generated by parabolic elements.

Contribution

It provides a recursive algorithm for determining shape parameters of 2-bridge links and applies triangulation methods to study Kleinian groups and their parameter spaces.

Findings

Explicit computation of fundamental domains and holonomies.

Determination of cusp fields for 2-bridge links.

Triangulations of Heckoid orbifolds.

Abstract

This paper develops a computational framework for studying the hyperbolic geometry of 2-bridge link complements and Kleinian groups generated by two parabolic elements. The framework is built on Sakuma-Weeks triangulations and introduces a family of Farey recursive polynomials. For a rational number which determines a hyperbolic 2-bridge link, this paper provides a simple recursive algorithm to determine a Farey recursive polynomial which has a root which determines the geometry of the link complement. This root is the shape parameter for a pair of tetrahedra in the Sakuma-Weeks triangulation. Using the same polynomials, the paper defines rational functions, one function for each edge in the Farey graph. Evaluating these rational functions at the root yields the complete collection of shape parameters for the triangulation. The Riley slice and its exterior lie in the complex plane and are traditionally viewed as parameter spaces for two-parabolic generated subgroups of PSL(2,C). This paper's triangulation-based approach provides a more geometric perspective to these parameter spaces. The Farey recursive methods apply throughout the Riley slice and its exterior, enabling computations for quotient orbifolds of Kleinian groups generated by parabolic pairs, incomplete hyperbolic spaces with non-discrete parameter groups, and others. Explicit calculations include the cusp groups in the boundary of the Riley slice and singly augmented 2-bridge link complements. Applications include: explicit computation of fundamental domains and holonomies for 2-bridge link complements; a precise correspondence between group elements and crossing circles in tangle diagrams; determination of cusp fields for 2-bridge links; geometric analysis of algebraic and geometric limits arising from Dehn surgery on singly augmented 2-bridge links; and explicit triangulations of Heckoid orbifolds.