Length functions of Hitchin representations

TL;DR

This paper constructs continuous length functions for Hitchin representations that generalize Thurston's length function, explores their properties, and applies them to eigenvalue estimates in higher rank settings.

Contribution

It introduces new length functions for Hitchin representations on geodesic currents, extending Thurston's classical length function to higher rank Lie groups.

Findings

Defined continuous length functions ^ ho for Hitchin representations

Established differentiability and identities for these length functions

Applied length functions to derive eigenvalue estimates

Abstract

Given a Hitchin representation $\rho \colon \pi_1(S) \to \PSL_n(\mathbb{R})$, we construct $n$ continuous functions $\ell_i^\rho \colon \mathcal \CH(S) \to \mathbb{R}$ defined on the space of H\"older geodesic currents $\CH(S)$ such that, for a closed, oriented curve $\gamma$ in $S$, the $i$--th eigenvalue of the matrix $\rho(\gamma)\in \PSL_n(\mathbb{R})$ is of the form $\pm \mathrm{exp}\, \ell_i^\rho(\gamma)$: such functions generalize to higher rank Thurston's length function of Fuchsian re\presentations. Identities, diffe\rentiability properties of these lengths $\ell_i^\rho$, as well as applications to eigenvalue estimates, are also considered.