On the Chabauty Space of PSL₂(ℝ), I: Lattices and Grafting

This is the first of two papers on the global topology of the space S u b (G) Sub (G) of all closed subgroups of G = P S L 2 (R) G=PSL₂ (R), equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of G 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 G, following Baik–Clavier. Then for a fixed finite type hyperbolizable 2 2 -orbifold S S, we show that the space S u b S (G) SubS (G) of all lattices Γ > G > G with Γ ∖ H 2 ≅ S H² S is a fiber orbibundle over the moduli space M (S) M (S). We describe the closure S u b S (G) ¯