Random Kleinian Groups, II : Two parabolic generators

TL;DR

This paper investigates the probability that two randomly chosen parabolic M"obius transformations generate a discrete Fuchsian or Kleinian group, providing precise probabilities, estimates, and computational bounds for these cases.

Contribution

It advances the understanding of random two-generator groups by precisely calculating the probability of discreteness for groups generated by parabolic elements and supporting findings with computational analysis.

Findings

Exact probability for discreteness of Fuchsian groups with two parabolic generators

Estimates and bounds for Kleinian groups generated by random parabolic elements

Computational analysis of the Riley slice under Bowditch's condition

Abstract

In earlier work we introduced geometrically natural probability measures on the group of all M\"obius transformations in order to study "random" groups of M\"obius transformations, random surfaces, and in particular random two-generator groups, that is groups where the generators are selected randomly, with a view to estimating the likely-hood that such groups are discrete and then to make calculations of the expectation of their associated parameters, geometry and topology. In this paper we continue that study and identify the precise probability that a Fuchsian group generated by two parabolic M\"obius transformations is discrete, and give estimates for the case of Kleinian groups generated by a pair of random parabolic elements which we support with a computational investigation into of the Riley slice as identified by Bowditch's condition, and establish rigorous bounds.