Comptes Rendus
no. 9
Differential Geometry/Group Theory
Growth of discrete groups of isometries in negative curvature: a gap-property
[Croissance des groupes discrets d'isométries en courbure strictement négative : un minorant universel]

Présenté par : Étienne Ghys

Gérard Besson1 ; Gilles Courtois2 ; Sylvestre Gallot1
1 Institut Fourier, université de Grenoble I, UMR 5582 CNRS-UJF, 38402 Saint-Martin-d'Hères, France
2 École polytechnique, centre de mathématiques, UMR 7640 du CNRS, 91128 Palaiseau cedex, France
Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 567-572.

Nous prouvons qu'un sous groupe de type fini Γ, non virtuellement nilpotent, du groupe des isométries d'une variété de Cartan–Hadamard de dimension n et de courbure sectionnelle vérifiant −a2⩽K⩽−1 est d'entropie algébrique minorée, Ent(Γ)⩾C(n,a)>0.

We prove that a finitely generated group acting without fixed point on a n-dimensional Cartan–Hadamard manifold of pinched sectional curvature −a2⩽K⩽−1 is either virtually nilpotent or has entropy Ent(Γ)⩾C(n,a)>0.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.09.025
Gérard Besson 1 ; Gilles Courtois 2 ; Sylvestre Gallot 1

1 Institut Fourier, université de Grenoble I, UMR 5582 CNRS-UJF, 38402 Saint-Martin-d'Hères, France
2 École polytechnique, centre de mathématiques, UMR 7640 du CNRS, 91128 Palaiseau cedex, France 
@article{CRMATH_2005__341_9_567_0,
     author = {G\'erard Besson and Gilles Courtois and Sylvestre Gallot},
     title = {Growth of discrete groups of isometries in negative curvature: a gap-property},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {567--572},
     publisher = {Elsevier},
     volume = {341},
     number = {9},
     year = {2005},
     doi = {10.1016/j.crma.2005.09.025},
     language = {en},
}
TY  - JOUR
AU  - Gérard Besson
AU  - Gilles Courtois
AU  - Sylvestre Gallot
TI  - Growth of discrete groups of isometries in negative curvature: a gap-property
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 567
EP  - 572
VL  - 341
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2005.09.025
LA  - en
ID  - CRMATH_2005__341_9_567_0
ER  - 
%0 Journal Article
%A Gérard Besson
%A Gilles Courtois
%A Sylvestre Gallot
%T Growth of discrete groups of isometries in negative curvature: a gap-property
%J Comptes Rendus. Mathématique
%D 2005
%P 567-572
%V 341
%N 9
%I Elsevier
%R 10.1016/j.crma.2005.09.025
%G en
%F CRMATH_2005__341_9_567_0
Gérard Besson; Gilles Courtois; Sylvestre Gallot. Growth of discrete groups of isometries in negative curvature: a gap-property. Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 567-572. doi : 10.1016/j.crma.2005.09.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.025/

[1] R.C. Alperin; G.A. Noskov Non vanishing of algebraic entropy for geometrically finite groups of isometries of Hadamard manifolds (Preprint) | arXiv

[2] G. Besson; G. Courtois; S. Gallot Rigidity of amalgamated products in negative curvature (Preprint) | arXiv

[3] G. Besson; G. Courtois; S. Gallot Un lemme de Margulis sans courbure et ses applications, Prépublication Inst. Fourier, Volume 595 (2003), pp. 1-59

[4] E. Breuillard; T. Gelander An effective Tits alternative http://www.dma.ens.fr/~breuilla/ (Abstract)

[5] E. Breuillard, T. Gelander, Cheeger constant and algebraic entropy of linear groups, Preprint, June 2005

[6] Y.D. Burago; V.A. Zalgaller Geometric Inequalities, Grundlehren Math. Wiss., vol. 285, Springer-Verlag, Berlin/New York, 1988

[7] P. De la Harpe Topics in Geometric Group Theory, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 2000

[8] P. De la Harpe Uniform growth in groups of exponential growth, Geom. Dedicata, Volume 95 (2002), pp. 1-17

[9] P. De la Harpe; R. Grigorchuk Limit behaviour of exponential growth rates for finitely generated groups, Monographie de l'Enseignement Mathématique, Volume 38 (2001), pp. 351-370

[10] A. Eskin; S. Mozes; H. Oh On uniform exponential growth for linear groups, Invent. Math., Volume 160 (2005), pp. 1-30

[11] M. Gromov Metric Structures for Riemannian and Non-Riemannian Spaces (J. Lafontaine; P. Pansu, eds.), Progress in Math., Birkhäuser, Basel, 1999

[12] M. Kapovitch, On Ping–Pong argument, Private communication, 2003 July 26th

[13] M. Koubi Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier, Volume 48 (1998), pp. 1441-1453

[14] D. Osin The entropy of solvable groups, Ergodic Theory Dynam. Systems, Volume 23 (2003), pp. 907-918

[15] J. Wilson On exponential growth and uniformly exponential growth for groups, Invent. Math., Volume 155 (2004), pp. 287-303

[16] X. Xie Growth of relatively hyperbolic groups (Preprint) | arXiv

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Some finiteness results for groups with bounded algebraic entropy

Fabio Zuddas

C. R. Math (2004)

Comparaison des volumes des variétés riemanniennes

Hamid-Reza Fanaï

C. R. Math (2004)

Hyperbolic manifolds, amalgamated products and critical exponents

Gérard Besson; Gilles Courtois; Sylvestre Gallot

C. R. Math (2003)