## Abstract

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}, SL_{2}(ℂ)), the SL_{2}(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}, SL_{2}(ℂ)) to a toric variety X(P_{r}). We relate X(F_{g}, SL_{2}(ℂ)) and X(v) topologically by showing that there is a surjective, continuous, proper map Ξ_{v} X(F_{g}, SL_{2}(ℂ)) → X(v). We then show that this map is a symplectomorphism on a dense open subset of X(F_{g}, SL_{2}(ℂ)) with respect to natural symplectic structures on X(F_{g}, SL_{2}(ℂ)) and X(v). In this way, we construct an integrable Hamiltonian system in X(F_{g}, SL_{2}(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}, SL_{2}(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}, SL_{2}(ℂ)).

Original language | English |
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Pages (from-to) | 354-399 |

Number of pages | 46 |

Journal | Canadian Journal of Mathematics |

Volume | 70 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2018 |

### Bibliographical note

Publisher Copyright:© Canadian Mathematical Society 2017.

## Keywords

- Analytification
- Character variety
- Compactification
- Integrable system
- Outer space

## ASJC Scopus subject areas

- Mathematics (all)

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