Toric Geometry of SL 2(C) Free Group Character Varieties from Outer Space

Manon, Christopher

Canadian journal of mathematics, Tome 70 (2018) no. 2, pp. 354-399

Résumé

Culler and Vogtmann defined a simplicial space $O\left( g \right)$ , called outer space, to study the outer automorphism group of the free group ${{F}_{g}}$ . Using representation theoretic methods, we give an embedding of $O\left( g \right)$ into the analytification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ , the $S{{L}_{2}}\left( \mathbb{C} \right)$ character variety of ${{F}_{g}}$ , reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O\left( g \right)$ defines a flat degeneration of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ to a toric variety $X\left( {{P}_{\Gamma }} \right)$ . We relate $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$ topologically by showing that there is a surjective, continuous, proper map ${{\Xi }_{v}}\,:\,x\left( {{F}_{g}}\,,\,S{{L}_{2}}\,\left( \mathbb{C} \right) \right)\,\to \,X\left( v \right)$ . We then show that this map is a symplectomorphism on a dense open subset of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ with respect to natural symplectic structures on $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$ . In this way, we construct an integrable Hamiltonian system in $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ for each point in a maximal cell of $O\left( g \right)$ , and we show that each $v$ defines a topological decomposition of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ derived from the decomposition of $X\left( {{P}_{\Gamma }} \right)$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O\left( g \right)$ all arise as divisorial valuations built from an associated projective compactification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ .

Détail

DOI : 10.4153/CJM-2016-042-0

Mots-clés : 14M25, 14T05, 14D20, character variety, outer space, analytification, compactification, integrable system