The structure of two-parabolic space: parabolic dust and iteration

Abstract

A non-elementary Möbius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space, the parameter space for representations of groups generated by two parabolics into $${PSL(2, \mathbb{C})}$$ modulo conjugacy, has been extensively studied. In this paper, we use the results of Gilman and Waterman to obtain an additional structure for the parameter space, which we term the two-parabolic space. This structure allows us to identify groups that contain additional conjugacy classes of primitive parabolics, which we call parabolic dust groups, non-free groups off the real axis, and groups that are both parabolic dust and non-free; some of these contain $${\mathbb{Z} \times \mathbb{Z}}$$ subgroups. The structure theorem also attaches additional geometric structure to discrete and non-discrete groups lying in given regions of the parameter space and allows a new explicit geometric construction of some non-classical T-Schottky groups.