Toric geometry of SL₂(ℂ) free group character varieties from outer space

Culler and Vogtmann defined a simplicial space O(g), 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(g) into the analytification of X(F_g, SL₂(ℂ)), the SL₂(ℂ) character variety of F_g, reproving a result of Morgan and Shalen. Then we show that every point v contained in a max-imal cell of O(g) defines a flat degeneration of X(F_g, SL₂(ℂ)) to a toric variety X(P_r). We relate X(F_g, SL₂(ℂ)) and X(v) topologically by showing that there is a surjective, continuous, proper map Ξ_v X(F_g, SL₂(ℂ)) → X(v). We then show that this map is a symplectomorphism on a dense open subset of X(F_g, SL₂(ℂ)) with respect to natural symplectic structures on X(F_g, SL₂(ℂ)) and X(v). In this way, we construct an integrable Hamiltonian system in X(F_g, SL₂(ℂ)) for each point in a maximal cell of O( g), and we show that each v deines a topological decomposition of X(F_g, SL₂(ℂ)) derived from the decomposition of X(P_r ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in O( g) all arise as divisorial valuations built from an associated projective compactification of X(F_g, SL₂(ℂ)).