Random Kleinian Groups, II Two Parabolic Generators

In earlier work, we introduced geometrically natural probability measures on the group of all Mobius transformations in order to study "random" groups of Mobius 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 likelihood 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 Mobius 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 of the Riley slice as identified by Bowditch's condition, and establish rigorous bounds.