Positive crossratios, barycenters, trees and applications to maximal representations (Q6594550)

Let \(S\) be a hyperbolic surface of finite area. This paper studies \textit{maximal representations} from \(\pi_1(S)\) to \(\mathrm{Sp}(2n,\mathbb{F})\) where \(\mathbb{F}\) is a real closed field. Such representations appear in the real spectrum compactification of the character variety as degenerations of real maximal representations (see [\textit{M. Burger} et al., C. R., Math., Acad. Sci. Paris 359, No. 4, 439--463 (2021; Zbl 1541.32002)]).\N\NLet \(\mathbb{F}\) be a real closed field which admits an order compatible valuation and let \[\rho:\pi_1(S)\longrightarrow \mathrm{Sp}(2n,\mathbb{F})\] be a maximal representation. The main result of this article is the existence of a geodesic current whose intersection with a closed geodesic \(\gamma\) on \(S\) equals the translation length of \(\rho(\gamma)\) (for the action on the building of \(\mathrm{Sp}(2n,\mathbb{F})\)). When \(\mathbb{F}=\mathbb{R}\), this result was shown by \textit{G. Martone} and \textit{T. Zhang} [Comment. Math. Helv. 94, No. 2, 273--345 (2019; Zbl 1422.37018)].\N\NThe strategy consists of associating to any maximal representation a \textit{positive cross-ratio}. Given a \(\pi_1(S)\)-invariant subset \(X\subset \partial\pi_1(S)\), a positive cross-ratio is a map \(B\) which associates a non-negative number to any positively ordered quadruple of points in \(X\), which is invariant under the action of \(\pi_1(S)\) and satisfies some symmetry conditions. To construct \(B\), the key point is that maximal representations admit an equivariant map from the subset of \(\partial \pi_1(S)\) consisting of fixed point of hyperbolic elements to the set of Lagrangians of \(\mathbb{F}^n\).\N\NThe above cross-ratio is constructed so that the translation length of a closed geodesic is equal to its \emph{period}. Let \(\gamma\) be a closed geodesic on \(S\) such that the endpoints \(\gamma^{\pm}\) of a lift of \(\gamma\) to the universal cover of \(S\) belong to \(X\). The period of \(\gamma\) for \(B\) is defined as \N\[\N\mathrm{Per}(\gamma)=B(\gamma^{-},x,\gamma x,\gamma^{+}),\N\]\Nwhere \(x\) is any point in \(X\) that makes \((\gamma^-,x,\gamma x,\gamma^{+})\) positively ordered. The authors finally show that for every positive cross-ratio \(B\), there exists a geodesic current whose intersection with a closed geodesic \(\gamma\) on \(S\) equals the period of \(\gamma\) for \(B\).\N\NThe authors also state various results concerning the nature of the geodesic current associated to a maximal representation \(\rho\). They show that if the image of the field generated by the matrix coefficients of \(\rho\) under the valuation is discrete in \(\mathbb{R}\), then the associated geodesic current is a multiple of a multi curve. Examples where this condition is satisfied are given at the end of the paper.