Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,R)

Abstract

This article studies the geometry of proper open convex domains in the projective space $\mathbb{R}{\mathbf{P}}^n$. These domains carry several projective invariant distances, among which are the Hilbert distance $d^H$ and the Blaschke distance $d^B$. We prove a thin inequality between those distances: for any two points $x$ and $y$ in such a domain,

$$d^B(x,y)<d^H(x,y)+1.$$

We then give two interesting consequences. The first one answers a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in $\mathbb{R}{\mathbf{P}}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation $\rho$ of a surface group $\Gamma$ into $PSL(3,\mathbb{R})$, there exists a Fuchsian representation $j:\Gamma \to PSL(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than that of $\rho$. This answers positively a conjecture of Lee and Zhang in the $3$-dimensional case.