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Bounding deformation spaces of Kleinian groups with two generators

A. Elzenaar, J. Gong, G. J. Martin, J. Schillewaert

TL;DR

This work establishes sharp, computable bounds on the deformation space of two-generator Kleinian groups ${\mathbb Z}_p*{\mathbb Z}_q$ by introducing a complex parameter ${\rho}$ and a polyhedral region ${\Omega}_{p,q}$ in which discreteness implies a free product structure. Using isometric discs, conic interactive pairs, and alternative parameterizations of the character variety, the authors derive explicit criteria ensuring discreteness and freeness, including precise inequalities and cusp-bound behavior that align with the highly fractal boundary of the deformation space. They apply these bounds to demonstrate a strengthened version of a conjecture on the faithfulness of specialized Burau representations of the braid group on three strands, linking hyperbolic geometry with quantum and knot invariants. The results provide practical, sharp tools for computer-assisted exploration of extremal Kleinian groups and their geometric/topological implications for hyperbolic 3-orbifolds, while revealing deep connections between quasiconformal deformation theory and quantum invariants.

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.

Bounding deformation spaces of Kleinian groups with two generators

TL;DR

This work establishes sharp, computable bounds on the deformation space of two-generator Kleinian groups by introducing a complex parameter and a polyhedral region in which discreteness implies a free product structure. Using isometric discs, conic interactive pairs, and alternative parameterizations of the character variety, the authors derive explicit criteria ensuring discreteness and freeness, including precise inequalities and cusp-bound behavior that align with the highly fractal boundary of the deformation space. They apply these bounds to demonstrate a strengthened version of a conjecture on the faithfulness of specialized Burau representations of the braid group on three strands, linking hyperbolic geometry with quantum and knot invariants. The results provide practical, sharp tools for computer-assisted exploration of extremal Kleinian groups and their geometric/topological implications for hyperbolic 3-orbifolds, while revealing deep connections between quasiconformal deformation theory and quantum invariants.

Abstract

In this article we provide simple and provable bounds on the size and shape of the locus of discrete subgroups of which split as a free product of cyclic groups , . 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 -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.
Paper Structure (8 sections, 17 theorems, 47 equations, 10 figures)

This paper contains 8 sections, 17 theorems, 47 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation and elementary definitions
  3. Conic interactive pairs
  4. Bounds using isometric circles
  5. First bounds via isometric discs
  6. The case that $A$ and $B$ are elliptic
  7. A different parameterisation of the character variety
  8. Faithful representations of the braid group on three strands

Key Result

Theorem 1.1

Let $X,Y\in \mathsf{SL}(2,\mathbb{C})$ be primitive elliptic elements of order $p$ and $q$, where $3\leq p\leq q$. Let $\rho$ be a solution of If $\rho\not\in \Omega_{p,q}$, then $\langle X,Y\rangle\cong \langle X\rangle*\langle Y\rangle$ and the group is discrete. This result is sharp at four cusp groups lying on the boundary of $\Omega_{p,q}$.

Figures (10)

  • Figure 1: The exterior of the illustrated regions is a concrete realisation of the quasiconformal deformation spaces $\overline{\mathcal{R}}_{3,3}$ (left) and $\overline{\mathcal{R}}_{4,4}$. The identified boundary points are certain "cusp"-groups of slope $\pm \frac{1}{n}$.
  • Figure 2: The shaded hexagonal region contains the exterior of the set of faithful discrete representations of ${\mathbb{Z}}_4*{\mathbb{Z}}_4$. It meets the boundary of the set at the cusp groups with slopes $\frac{0}{1}$, $\pm \frac{1}{2}$, and $\frac{1}{1}$.
  • Figure 3: Stellations of the regular heptagon the and square.
  • Figure 4: The translates of the sector $\mathbf{K}/\mu$ line up and are disjoint. The sector with apex $\mathbf{K}/\mu$ maps onto the sector in the top left under rotation by $A$.
  • Figure 5: The new fundamental domain for $A$ consists of $\mathbf{K}$ together with the isometric discs of $Y$ and suitable images deleted. Left: The real part of the centres of discs have opposite signs. Middle: Real part of centres have same (negative) sign. Right: Elliptic example.
  • ...and 5 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • ...and 24 more