The space of geometric limits of one-generator closed subgroups of $\mathrm{PSL}_2(\mathbb{R})$

Abstract

We give a complete description of the closure of the space of one-generator closed subgroups of ${PSL}_2\left(ℝ\right)$ for the Chabauty topology, by computing explicitly the matrices associated with elements of $Aut\left(\mathbb{D}\right)\cong {PSL}_2\left(ℝ\right)$, and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps ${\phi }_n:X\to Y$ transform convergent sequences of closed subsets of the domain $X$ into convergent sequences of closed subsets of the range $Y$. In particular, this allows us to compute certain geometric limits of ${PSL}_2\left(ℝ\right)$ only by looking at the Hausdorff limit of some closed subsets of $ℂ$.