On the Hilbert geometry of simplicial Tits sets
[Sur la géométrie de Hilbert d’ensembles de Tits simpliciaux]
Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1005-1030.

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.

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.

DOI : https://doi.org/10.5802/aif.2950
Classification : 20F67,  51F15,  53C60
Mots clés : structure projective convexe, groupe de réflexion, géométrie de Hilbert, entropie volumique
@article{AIF_2015__65_3_1005_0,
     author = {Nie, Xin},
     title = {On the {Hilbert} geometry of simplicial {Tits} sets},
     journal = {Annales de l'Institut Fourier},
     pages = {1005--1030},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     doi = {10.5802/aif.2950},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2950/}
}
TY  - JOUR
AU  - Nie, Xin
TI  - On the Hilbert geometry of simplicial Tits sets
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 1005
EP  - 1030
VL  - 65
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2950/
UR  - https://doi.org/10.5802/aif.2950
DO  - 10.5802/aif.2950
LA  - en
ID  - AIF_2015__65_3_1005_0
ER  - 
Nie, Xin. On the Hilbert geometry of simplicial Tits sets. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1005-1030. doi : 10.5802/aif.2950. http://archive.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 2094116 | Zbl 1084.37026

[2] Benoist, Yves Convexes divisibles. III, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 5, pp. 793-832 | Article | Numdam | MR 2195260 | Zbl 1085.22006

[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 2655311 | Zbl 1252.22006

[4] Crampon, Mickaël Entropies of strictly convex projective manifolds, J. Mod. Dyn., Volume 3 (2009) no. 4, pp. 511-547 | Article | MR 2587084 | Zbl 1189.37034

[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) | Article | MR 1086648

[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 | Article | MR 957518 | Zbl 0659.57004

[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 1053346 | Zbl 0711.53033

[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 | Article | MR 1238518 | Zbl 0832.52002

[9] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, pp. xii+204 | MR 1066460 | Zbl 0768.20016

[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 42129 | Zbl 0037.39802

[11] Manning, Anthony Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | Article | MR 554385 | Zbl 0426.58016

[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 1748082 | Zbl 0958.22008

[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 283720 | Zbl 0206.51302

[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) | Article

[15] Vinberg, Ernest; Kac, Victor Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR 208470 | Zbl 0163.16902

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

Cité par Sources :