Outer space and the combinatorics of character varieties

Thursday, March 26, 2015 -
3:00pm to 4:00pm
For a discrete group $\pi$ and a reductive group $G$, the character variety $\mathcal{X}(\pi, G)$ is the moduli space of semi-stable representations of $\pi$ into $G.$ Many moduli spaces appear in the guise of character varieties, for example for a manifold $M$ with fundamental group $\pi_1(M) = \pi$, $\mathcal{X}(\pi, G)$ is the space of flat topological $G-$bundles on $M$, and if $M$ is equipped with a K\"ahler structure, $\mathcal{X}(\pi, G)$ is the space of $G-$Higgs bundles. We will discuss several related combinatorial and geometric features of the character variety $\mathcal{X}(F_g, SL_2(C))$, where $F_g$ is the free group on $g$ generators and $G = SL_2(C)$ is the special linear group of $2\times2$ matrices. For every choice of graph $\Gamma$ equipped with an isomorphism $\gamma: \pi_1(\Gamma) \cong F_g$ we describe a so-called Newton-Okounkov body for $\mathcal{X}(F_g, SL_2(C))$, this is a convex cone which informs the algebraic structure of the coordinate ring $C[\mathcal{X}(F_g, SL_2(C))]$. We also discuss a related integrable system in $\mathcal{X}(F_g, SL_2(C)$, and a polyhedral cone of discrete valuations on $C[\mathcal{X}(F_g, SL_2(C))]$ for each choice of graph $(\Gamma, \gamma).$ We conclude by describing how these polyhedra knit together to give a map from Culler and Vogtman's "outer space" into the Berkovich analytification of $\mathcal{X}(F_g, SL_2(C))$, giving a new proof of a result of Morgan and Shalen.
Christopher Manon
George Mason University
Event Location: 
Fine Hall 314