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

Nicolas Tholozan (1, 2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 Uni.lu - Université du Luxembourg

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.

Domaines: Mathématiques [math]; Géométrie métrique [math.MG]; Géométrie différentielle [math.DG]; Systèmes dynamiques [math.DS]; Topologie géométrique [math.GT]

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⟩