Abstract
This article studies the geometry of proper open convex domains in the projective space . These domains carry several projective invariant distances, among which are the Hilbert distance and the Blaschke distance . We prove a thin inequality between those distances: for any two points and in such a domain,
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 , the volume of a ball of radius grows at most like . The second consequence is the following fact: for any Hitchin representation of a surface group into , there exists a Fuchsian representation such that the length spectrum of is uniformly smaller than that of . This answers positively a conjecture of Lee and Zhang in the -dimensional case.
Citation
Nicolas Tholozan. "Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into ." Duke Math. J. 166 (7) 1377 - 1403, 15 May 2017. https://doi.org/10.1215/00127094-00000010X
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