On the Hilbert geometry of simplicial Tits sets

Nie, Xin

Annales de l'Institut Fourier, Tome 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 $0$.

Classification : 20F67, 51F15, 53C60
Keywords: convex projective structure, reflection group, Hilbert geometry, volume entropy
Mots-clés : structure projective convexe, groupe de réflexion, géométrie de Hilbert, entropie volumique

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

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

[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 | Zbl | MR

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

[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 | Zbl | MR

[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/... | Zbl | MR

[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 | Zbl | MR

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

[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 | Zbl | MR

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

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

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

[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 | Zbl | MR

[16] Zhang, Tengren The degeneration of convex ${\mathrm{ℝ\mathbb{P}}}^2$ structures on surfaces (http://arxiv.org/abs/1312.2452)