Bounding deformation spaces of Kleinian groups with two generators

2024/05/24 by Elzenaar, A., Gong, J., Martin, G. J. +1 · 2 citations
#05A30 #20F65 #30F40 #32G15 #57K32 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.15970

Abstract

In this article we provide simple and provable bounds on the size and shape of the locus of discrete subgroups of PSL(2,ℂ)≅ Isom⁺(ℍ³) which split as a free product of cyclic groups ℤ_p*ℤ_q, 3≤ p,q ≤ ∞. 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.

Cited by

• On q-real and q-complex numbers
• Peripheral subgroups of Kleinian groups

Related

• The Geometry and Arithmetic of Kleinian Groups
• Constructing geometrically infinite groups on boundaries of deformation spaces
• Peripheral subgroups of Kleinian groups
• Strong convergence of Kleinian groups: the cracked eggshell
• The topology of deformation spaces of Kleinian groups