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

<p> This is the first of two papers on the global topology of the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S u b left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>u</mml:mi> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Sub(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of all closed subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals upper P upper S upper L 2 left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G=PSL_2(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , equipped with the Chabauty topology. In this paper, we study the spaces of lattices and elementary subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and prove a continuity result for conformal grafting of (possibly infinite type) vectored orbifolds that will be useful in both papers. </p> <p> More specifically, we first identify the homotopy type of the space of elementary subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , following Baik–Clavier. Then for a fixed finite type hyperbolizable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -orbifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we show that the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S u b Subscript upper S Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Sub_S(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of all lattices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma greater-than upper G"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo>></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma > G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma minus double-struck upper H squared approximately-equals upper S"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mi class="MJX-variant" mathvariant="normal"> ∖ </mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> ≅ </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \backslash \mathbb {H}^2 \cong S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a fiber orbibundle over the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal M(S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We describe the closure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper S u b Subscript upper S Baseline left-parenthesis upper G right-parenthesis With bar"> <mml:semantics> <mml:mover> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo>