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Toric Geometry of SL2(ℂ) Free Group Character Varieties from Outer Space

Published online by Cambridge University Press:  20 November 2018

Christopher Manon
Christopher Manon*
Affiliation:
Department of Mathematics, George Mason University, Fairfax, VA 22030 USA e-mail: cmanon@gmu.edu
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Abstract

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Culler and Vogtmann defined a simplicial space $O\left( g \right)$, called outer space, to study the outer automorphism group of the free group ${{F}_{g}}$. Using representation theoretic methods, we give an embedding of $O\left( g \right)$ into the analytification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$, the $S{{L}_{2}}\left( \mathbb{C} \right)$ character variety of ${{F}_{g}}$, reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O\left( g \right)$ defines a flat degeneration of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ to a toric variety $X\left( {{P}_{\Gamma }} \right)$. We relate $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$ topologically by showing that there is a surjective, continuous, proper map ${{\Xi }_{v}}\,:\,x\left( {{F}_{g}}\,,\,S{{L}_{2}}\,\left( \mathbb{C} \right) \right)\,\to \,X\left( v \right)$. We then show that this map is a symplectomorphism on a dense open subset of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ with respect to natural symplectic structures on $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ and $X\left( v \right)$. In this way, we construct an integrable Hamiltonian system in $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ for each point in a maximal cell of $O\left( g \right)$, and we show that each $v$ defines a topological decomposition of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$ derived from the decomposition of $X\left( {{P}_{\Gamma }} \right)$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O\left( g \right)$ all arise as divisorial valuations built from an associated projective compactification of $x\left( {{F}_{g}}\,,\,S{{L}_{2}}\left( \mathbb{C} \right) \right)$.

Keywords

14M2514T0514D20character varietyouter spaceanalytificationcompactificationintegrable system
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 2 , 01 April 2018 , pp. 354 - 399
Copyright
Copyright © Canadian Mathematical Society 2018

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