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 RP n. These domains carry several projective invariant distances, among which 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 to a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in RP 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 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 to a conjecture of Lee and Zhang in the three-dimensional case.
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Soumis le : mardi 15 mai 2018 - 16:13:44
Dernière modification le : mercredi 10 octobre 2018 - 10:09:09
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  • HAL Id : hal-01792658, version 1

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Nicolas Tholozan. Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3,R) . 2018. 〈hal-01792658〉

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