Projections in Moduli Spaces of the Kleinian Groups

Abstract

A two-generator Kleinian group $〈f,g〉$ can be naturally associated with a discrete group $〈f,\varphi 〉$ with the generator $\varphi$ of order two and where $〈f,\varphi f{\varphi }^{-1}〉=〈f,gf{g}^{-1}〉\subset 〈f,g〉,\left[〈f,\varphi 〉\text{:}〈f,gf{g}^{-1}〉\right]=2.$ This is useful in studying the geometry of the Kleinian groups since $〈f,g〉$ will be discrete only if $〈f,\varphi 〉$ is, and the moduli space of groups $〈f,\varphi 〉$ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of $〈f,g〉$ . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.