AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).
Topological mirror symmetry for rank two character varieties of surface groups
