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

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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)).

Original languageEnglish
Pages (from-to)354-399
Number of pages46
JournalCanadian Journal of Mathematics
Issue number2
StatePublished - Apr 2018

Bibliographical note

Publisher Copyright:
© Canadian Mathematical Society 2017.


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

ASJC Scopus subject areas

  • Mathematics (all)


Dive into the research topics of 'Toric geometry of SL2(ℂ) free group character varieties from outer space'. Together they form a unique fingerprint.

Cite this