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Projections in moduli spaces of Kleinian groups

A two-generator Kleinian group $\langle f,g \rangle$ can be naturally associated with a discrete group $\langle f,ϕ\rangle$ with the generator $ϕ$ of order $2$ and where \begin{equation*} \langle f,ϕf ϕ^{-1} \rangle= \langle f,gfg^{-1} \rangle \subset \langle f,g\rangle, \quad [ \langle f,g f g^{-1} \rangle: \langle f,ϕ\rangle]=2 \end{equation*} This is useful in studying the geometry of Kleinian groups since $\langle f,g \rangle$ will be discrete only if $\langle f,ϕ\rangle$ is, and the moduli space of groups $\langle f,ϕ\rangle$ 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 Jørgensen's results on algebraic convergence.