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Article Dans Une Revue Duke Mathematical Journal Année : 2017

## Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\mathrm{PSL}(3,\mathbb R)$

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Nicolas Tholozan
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#### Résumé

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.

### Dates et versions

hal-01792658 , version 1 (15-05-2018)

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• HAL Id : hal-01792658 , version 1

### Citer

Nicolas Tholozan. Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\mathrm{PSL}(3,\mathbb R)$. Duke Mathematical Journal, 2017, 166 (7), pp.1377-1403. ⟨hal-01792658⟩

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