Hitchin singled out a preferred component $\mathrm{Hit}_n(S)$ in the character variety of representations from the fundamental group of a surface $S$ to $\mathrm{PSL}_n(\mathbb R)$. When $n=2$, $\mathrm{Hit}_2(S)$ coincides with the Teichm\"uller space $\mathcal T(S)$ consisting of all hyperbolic metrics on the surface $S$. Later Labourie showed that the elements in $\mathrm{Hit}_n(S)$ share many important differential geometric and dynamical properties.
Morgan and Shalen provided an algebro-geometric interpretation of Thurston's compactification of $\mathcal T(S)$ in terms of valuations on character varieties. Parreau extended this construction to a compactification of $\mathrm{Hit}_n(S)$ whose boundary points are described by actions of $\pi_1(S)$ on an $\mathbb{R}$-building $\mathcal B$. This generalizes the actions on $\mathbb{R}$-trees occurring for the Morgan-Shalen compactification of $\mathcal T(S)$.
In this talk, we offer a new presentation for the Parreau compactification of $\mathrm{Hit}_n(S)$, which is based on certain positivity properties discovered by Fock and Goncharov. More precisely, we use the Fock-Goncharov construction to describe the intersection patterns of apartments in $\pi_1(S)$-invariant subsets of $\mathcal B$ that arise in the boundary of $\mathrm{Hit}_n(S)$. Speaker(s): Giuseppe Martone (University of Southern California)