The Structure of Two-parabolic Space: Parabolic Dust and Iteration

A non-elementary M\"obius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space has been extensively studied. In this paper, we use the results of \cite{GW} to obtain an additional structure for the parameter space, which we term the {\sl two-parabolic space}. This structure allows us to identify groups that contain additional conjugacy classes of primitive parabolics, which following \cite{Indra} we call {\sl 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 including a new explicit construction of some non-classical $\T$Schottky groups.