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 RPn. These domains carry several projective invariant distances, among which are the Hilbert distance dH and the Blaschke distance dB. We prove a thin inequality between those distances: for any two points x and y in such a domain, dB(x,y)<dH(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 RPn, 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 ρ of a surface group Γ into PSL(3,R), there exists a Fuchsian representation j:Γ→PSL(2,R) such that the length spectrum of j is uniformly smaller than that of ρ. This answers positively a conjecture of Lee and Zhang in the 3-dimensional case.