On the Hilbert geometry of simplicial Tits sets
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1005-1030.

The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy of the Hilbert metric tends to 0.

L’espace des modules de structures projectives convexes sur un orbifold simplicial hyperbolique est soit un point soit la droite réelle. En répondant à une question de M. Crampon, nous prouvons que dans ce dernier cas, quand on tend vers l’infini dans l’espace des modules, l’entropie de la métrique de Hilbert tend vers 0.

DOI: 10.5802/aif.2950
Classification: 20F67,  51F15,  53C60
Keywords: convex projective structure, reflection group, Hilbert geometry, volume entropy
Nie, Xin 1

1 Tsinghua University Dept. of Mathematics Beijing 100084 (China)
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Nie, Xin. On the Hilbert geometry of simplicial Tits sets. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1005-1030. doi : 10.5802/aif.2950. http://www.numdam.org/articles/10.5802/aif.2950/

[1] Benoist, Yves Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 339-374 | MR | Zbl

[2] Benoist, Yves Convexes divisibles. III, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 5, pp. 793-832 | DOI | Numdam | MR | Zbl

[3] Benoist, Yves Five lectures on lattices in semisimple Lie groups, Géométries à courbure négative ou nulle, groupes discrets et rigidités (Sémin. Congr.), Volume 18, Soc. Math. France, Paris, 2009, pp. 117-176 | MR | Zbl

[4] Crampon, Mickaël Entropies of strictly convex projective manifolds, J. Mod. Dyn., Volume 3 (2009) no. 4, pp. 511-547 | DOI | MR | Zbl

[5] Ghys, É.; de la Harpe, P. Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, 83, Birkhäuser Boston, Inc., Boston, MA, 1990, pp. xii+285 (Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988) | DOI | MR

[6] Goldman, William M. Geometric structures on manifolds and varieties of representations, Geometry of group representations (Boulder, CO, 1987) (Contemp. Math.), Volume 74, Amer. Math. Soc., Providence, RI, 1988, pp. 169-198 | DOI | MR | Zbl

[7] Goldman, William M. Convex real projective structures on compact surfaces, J. Differential Geom., Volume 31 (1990) no. 3, pp. 791-845 http://projecteuclid.org/getRecord?id=euclid.jdg/1214444635 | MR | Zbl

[8] de la Harpe, Pierre On Hilbert’s metric for simplices, Geometric group theory, Vol. 1 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.), Volume 181, Cambridge Univ. Press, Cambridge, 1993, pp. 97-119 | DOI | MR | Zbl

[9] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, pp. xii+204 | MR | Zbl

[10] Lannér, Folke On complexes with transitive groups of automorphisms, Comm. Sém., Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], Volume 11 (1950), pp. 71 | MR | Zbl

[11] Manning, Anthony Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | DOI | MR | Zbl

[12] Margulis, Grigory; Vinberg, Ernest Some linear groups virtually having a free quotient, J. Lie Theory, Volume 10 (2000) no. 1, pp. 171-180 | MR | Zbl

[13] Vey, Jacques Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa (3), Volume 24 (1970), pp. 641-665 | Numdam | MR | Zbl

[14] Vinberg, Ernest Geometry. II, Encyclopaedia of Mathematical Sciences, 29, Springer-Verlag, Berlin, 1993, pp. viii+254 (Spaces of constant curvature, A translation of Geometriya. II, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, Translation by V. Minachin [V. V. Minakhin], Translation edited by È. B. Vinberg) | DOI

[15] Vinberg, Ernest; Kac, Victor Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR | Zbl

[16] Zhang, Tengren The degeneration of convex ℝℙ 2 structures on surfaces (http://arxiv.org/abs/1312.2452)

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