Projections in moduli spaces of Kleinian groups

TL;DR

This paper explores a projection method in the moduli space of two-generator Kleinian groups, simplifying the analysis of their discreteness by reducing the problem to a lower-dimensional space and establishing the closedness of the image.

Contribution

It introduces a projection of principal characters that reduces the moduli space dimension and proves the image's closedness using Jørgensen's algebraic convergence results.

Findings

The projection of the moduli space is closed.

Discreteness of groups can be studied via simpler moduli spaces.

The method leverages Jørgensen's convergence theorems.

Abstract

A two-generator Kleinian group $\langle f,g \rangle$ can be naturally associated with a discrete group $\langle f,\phi \rangle$ with the generator $\phi$ of order $2$ and where \begin{equation*} \langle f,\phi f \phi^{-1} \rangle= \langle f,gfg^{-1} \rangle \subset \langle f,g\rangle, \quad [ \langle f,g f g^{-1} \rangle: \langle f,\phi \rangle]=2 \end{equation*} This is useful in studying the geometry of Kleinian groups since $\langle f,g \rangle$ will be discrete only if $\langle f,\phi \rangle$ is, and the moduli space of groups $\langle f,\phi \rangle$ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of $\langle f,g \rangle$. The dimension reduction here is realised by a projection of principal characters of 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 J\o rgensen's results on algebraic convergence.