Length asymptotics in higher Teichmüller theory (Q2862174)

Let \(S\) be a closed connected oriented surface of genus \(g \geq 2\) and \(\rho : \pi_1 (S) \rightarrow \mathrm{PSL}_n (\mathbb{R})\) a representation inside a Hitchin component. Such representations are faithful, discrete and for every non trivial element \(\alpha \in \pi_1 (S)\), the element \(\rho (\alpha)\) is diagonalizable with distinct real eigenvalues \(| \lambda_1 (\alpha)| < \dots < | \lambda_n (\alpha)| \), as shown by results of \textit{F. Labourie} [Invent. Math. 165, No. 1, 51--114 (2006; Zbl 1103.32007)]. Using this, we can define the width function by \(l_\rho (\alpha) = \mathrm{log} |\lambda_1 (\alpha)| - \mathrm{log} |\lambda_n (\alpha)|\).NEWLINENEWLINEIn this short article the authors are interested in counting problems and asymptotic behavior associated to these lengths and width. The main purpose of this article is to give a new proof of a result by \textit{A. Sambarino} [Quelques aspects des représentations linéaires des groupes hyperboliques. Paris: Université Paris 13 (PhD Thesis) (2011)] stating that there exists an analytic positive function \(\delta\) on the Hitchin component such that for all \(\rho\) : NEWLINE\[NEWLINE \# \{ \alpha \, : \, l_\rho (\alpha) \leq T \} \sim \dfrac{e^{\delta (\rho) T} }{\delta (\rho) T}, \text{ as } T \rightarrow + \infty NEWLINE\]NEWLINENEWLINENEWLINEThe proof is based on the dynamics of the geodesic flow on the bundle defined by a representation in the Hitchin component. The authors also use a recent construction of \textit{G. Dreyer} [Algebr. Geom. Topol. 13, No. 6, 3153--3173 (2013; Zbl 1285.57010)] relating the length functions to Hölder continuous functions, and a symbolic model of the geodesic flow.NEWLINENEWLINEFrom this main theorem, the authors deduce several distributional and limiting results such as equidistribution, large deviations and a central limit theorem.