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- W4300693741 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 $mathcal{X}(F_g, SL_2(mathbb{C})),$ the $SL_2(mathbb{C})$ 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 $O(g)$ defines a flat degeneration of $mathcal{X}(F_g, SL_2(mathbb{C}))$ to a toric variety $X(P_{Gamma})$. We relate $mathcal{X}(F_g, SL_2(mathbb{C}))$ and $X(v)$ topologically by showing that there is a surjective, continuous, proper map $Xi_v: mathcal{X}(F_g, SL_2(mathbb{C})) to X(v)$. We then show that this map is a symplectomorphism on a dense, open subset of $mathcal{X}(F_g, SL_2(mathbb{C}))$ with respect to natural symplectic structures on $mathcal{X}(F_g, SL_2(mathbb{C}))$ and $X(v)$. In this way, we construct an integrable Hamiltonian system in $mathcal{X}(F_g, SL_2(mathbb{C}))$ for each point in a maximal cell of $O(g)$, and we show that each $v$ defines a topological decomposition of $mathcal{X}(F_g, SL_2(mathbb{C}))$ derived from the decomposition of $X(v)$ 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 $mathcal{X}(F_g, SL_2(mathbb{C})).$" @default.
- W4300693741 created "2022-10-04" @default.
- W4300693741 creator A5069339031 @default.
- W4300693741 date "2014-09-30" @default.
- W4300693741 modified "2023-09-29" @default.
- W4300693741 title "Toric geometry of $SL_2(mathbb{C})$ free group character varieties from outer space" @default.
- W4300693741 doi "https://doi.org/10.48550/arxiv.1410.0072" @default.
- W4300693741 hasPublicationYear "2014" @default.
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