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(Fg, SL2(ℂ)), the SL2(ℂ) character variety of Fg, 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(Fg, SL2(ℂ)) to a toric variety X(Pr). We relate X(Fg, SL2(ℂ)) and X(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)) and X(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell of O( g), and we show that each v deines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition of X(Pr ) 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(Fg, SL2(ℂ)).
|Number of pages||46|
|Journal||Canadian Journal of Mathematics|
|State||Published - Apr 2018|
Bibliographical notePublisher Copyright:
© Canadian Mathematical Society 2017.
- Character variety
- Integrable system
- Outer space
ASJC Scopus subject areas
- Mathematics (all)