On the Chabauty space of $\textrm{PSL}_2(\mathbb{R})$, I: lattices and grafting

TL;DR

This paper explores the topology of the space of all closed subgroups of PSL(2,R), focusing on lattices and elementary subgroups, and introduces a continuity result for conformal grafting of vectored orbifolds, with implications for hyperbolic geometry.

Contribution

It characterizes the homotopy type of elementary subgroups, describes the topology of lattice spaces, and establishes a new convergence result for grafted hyperbolic orbifolds.

Findings

The space of elementary subgroups has a known homotopy type.

Lattice spaces form fiber orbibundles over moduli spaces.

Grafted orbifolds converge smoothly to the expected limit.

Abstract

This is the first of two papers on the global topology of the space $\textrm{Sub}(G)$ of all closed subgroups of $G=\textrm{PSL}_2(\mathbb{R})$, equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of $G$, and prove a continuity result for conformal grafting of (possibly infinite type) vectored orbifolds that will be useful in both papers. More specifically, we first identify the homotopy type of the space of elementary subgroups of $G$, following Baik-Clavier. Then for a fixed finite type hyperbolizable $2$-orbifold $S$, we show that the space $\textrm{Sub}_S(G)$ of all lattices $\Gamma < G$ with $\Gamma \backslash \mathbb{H}^2 \cong S$ is a fiber orbibundle over the moduli space $\mathcal M(S)$. We describe the closure $\overline{\textrm{Sub}_S(G)}$ in $\textrm{Sub}(G)$ and show that $\partial \textrm{Sub}_S(G)$ has a neighborhood deformation retract within $\overline{\textrm{Sub}_S(G)}$. When $S$ is not one of finitely many low complexity orbifolds, we show that $\overline{\textrm{Sub}_S(G)}$ is simply connected. In the simplest exceptional case, when $S$ is a sphere with three total cusps and cone points, we show that $\overline{\textrm{Sub}_S(G)}$ is a (usually nontrivial) lens space. Finally, we show that when $(X_i,v_i) \to (X_\infty,v_\infty)$ is a (possibly infinite type) smoothly converging sequence of vectored hyperbolic $2$-orbifolds, and we graft in Euclidean annuli along suitable collections of simple closed curves in the $X_i$, then after uniformization, the resulting vectored hyperbolic $2$-orbifolds converge smoothly to the expected limit. As part of the proof, we give a new lower bound on the hyperbolic distance between points in a grafted orbifold in terms of their original distance.