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 .
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 .
Keywords: convex projective structure, reflection group, Hilbert geometry, volume entropy
Mot clés : structure projective convexe, groupe de réflexion, géométrie de Hilbert, entropie volumique
Nie, Xin 1
@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 = {https://aif.centre-mersenne.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 SP - 1005 EP - 1030 VL - 65 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2950/ DO - 10.5802/aif.2950 LA - en ID - AIF_2015__65_3_1005_0 ER -
%0 Journal Article %A Nie, Xin %T On the Hilbert geometry of simplicial Tits sets %J Annales de l'Institut Fourier %D 2015 %P 1005-1030 %V 65 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2950/ %R 10.5802/aif.2950 %G en %F AIF_2015__65_3_1005_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2950/
[1] Convexes divisibles. I, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 339-374 | MR | Zbl
[2] Convexes divisibles. III, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 5, pp. 793-832 | DOI | Numdam | MR | Zbl
[3] 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] Entropies of strictly convex projective manifolds, J. Mod. Dyn., Volume 3 (2009) no. 4, pp. 511-547 | DOI | MR | Zbl
[5] 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] 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] 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] 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] Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, pp. xii+204 | MR | Zbl
[10] 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] Topological entropy for geodesic flows, Ann. of Math. (2), Volume 110 (1979) no. 3, pp. 567-573 | DOI | MR | Zbl
[12] Some linear groups virtually having a free quotient, J. Lie Theory, Volume 10 (2000) no. 1, pp. 171-180 | MR | Zbl
[13] Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa (3), Volume 24 (1970), pp. 641-665 | Numdam | MR | Zbl
[14] 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] Quasi-homogeneous cones, Mat. Zametki, Volume 1 (1967), pp. 347-354 | MR | Zbl
[16] The degeneration of convex structures on surfaces (http://arxiv.org/abs/1312.2452)
Cited by Sources: