Bounding deformation spaces of Kleinian groups with two generators

TL;DR

This paper establishes sharp bounds on the deformation spaces of two-generator Kleinian groups, aiding in the classification and analysis of hyperbolic 3-orbifolds and related algebraic representations.

Contribution

It provides provable bounds on the deformation space of certain Kleinian groups, with applications to quantum algebra and hyperbolic geometry.

Findings

Bounds are sharp and meet the fractal boundary of the deformation space.

Application to a conjecture on the faithfulness of the specialised Burau representation.

Facilitates computer-assisted searches for extremal Kleinian groups.

Abstract

In this article we provide simple and provable bounds on the size and shape of the locus of discrete subgroups of $\mathsf{PSL}(2,\mathbb{C})\cong \operatorname{Isom}^+(\mathbb{H}^3)$ which split as a free product of cyclic groups $\mathbb{Z}_p*\mathbb{Z}_q$, $3\leq p,q \leq \infty$. These bounds are sharp and meet the highly fractal boundary of the deformation space in four cusp groups. Such bounds have great utility in computer assisted searches for extremal Kleinian groups so as to identify universal constraints (volume, length spectra, etc.) on the geometry and topology of hyperbolic $3$-orbifolds. As an application, we prove a strengthened version of a conjecture by Morier-Genoud, Ovsienko, and Veselov, motivated by the theory of quantum rational numbers, on the faithfulness of the specialised Burau representation.