Group Theory/Probability Theory
Critical densities for random quotients of hyperbolic groups
[Densités critiques pour les quotients aléatoires de groupes hyperboliques]

Présenté par : Étienne Ghys

Ollivier, Yann1
1 Laboratoire de mathématique d'Orsay, UMR 8628 du CNRS, bâtiment 425, Université de Paris-Sud, 91405 Orsay, France
Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 391-394.

Nous prouvons que pour plusieurs modèles naturels de quotient aléatoire d'un groupe, dépendant d'un paramètre de densité, pour chaque groupe hyperbolique il existe une densité critique sous laquelle un quotient aléatoire reste hyperbolique avec grande probabilité, tandis qu'au-dessus de cette densité le quotient aléatoire est très probablement trivial. Nous donnons des caractérisations explicites de ces densités critiques dans les différents modèles.

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models.

Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00084-0
Ollivier, Yann 1

1 Laboratoire de mathématique d'Orsay, UMR 8628 du CNRS, bâtiment 425, Université de Paris-Sud, 91405 Orsay, France 
@article{CRMATH_2003__336_5_391_0,
     author = {Ollivier, Yann},
     title = {Critical densities for random quotients of hyperbolic groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--394},
     publisher = {Elsevier},
     volume = {336},
     number = {5},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00084-0},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00084-0/}
}
TY  - JOUR
AU  - Ollivier, Yann
TI  - Critical densities for random quotients of hyperbolic groups
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 391
EP  - 394
VL  - 336
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00084-0/
DO  - 10.1016/S1631-073X(03)00084-0
LA  - en
ID  - CRMATH_2003__336_5_391_0
ER  - 
%0 Journal Article
%A Ollivier, Yann
%T Critical densities for random quotients of hyperbolic groups
%J Comptes Rendus. Mathématique
%D 2003
%P 391-394
%V 336
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00084-0/
%R 10.1016/S1631-073X(03)00084-0
%G en
%F CRMATH_2003__336_5_391_0
Ollivier, Yann. Critical densities for random quotients of hyperbolic groups. Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 391-394. doi : 10.1016/S1631-073X(03)00084-0. http://www.numdam.org/articles/10.1016/S1631-073X(03)00084-0/

[1] Arzhantseva, G.N. Generic properties of finitely presented groups and Howson's theorem, Comm. Algebra, Volume 26 (1998) no. 4, pp. 3783-3792

[2] Arzhantseva, G.N.; Ol'shanskii, A.Yu. Generality of the class of groups in which subgroups with a lesser number of generators are free, Mat. Zametki, Volume 59 (1996) no. 4, pp. 489-496 (Translation in Math. Notes, 59, 3–4, 1996, pp. 350-355)

[3] Champetier, C. Cocroissance des groupes à petite simplification, Bull. London Math. Soc., Volume 25 (1993) no. 5, pp. 438-444

[4] Champetier, C. Propriétés statistiques des groupes de présentation finie, J. Adv. Math., Volume 116 (1995) no. 2, pp. 197-262

[5] Champetier, C. L'espace des groupes de type fini, Topology, Volume 39 (2000) no. 4, pp. 657-680

[6] Cohen, J.M. Cogrowth and amenability of discrete groups, J. Funct. Anal., Volume 48 (1982), pp. 301-309

[7] Ghys, É.; de la Harpe, P. Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., 83, Birkhäuser, 1990

[8] Grigorchuk, R.I. Symmetrical random walks on discrete groups (Dobrushin, R.L.; Sinai, Ya.G., eds.), Multicomponent Random Systems, Adv. Probab. Related Topics, 6, Dekker, 1980, pp. 285-325

[9] Gromov, M. Hyperbolic groups (Gersten, S.M., ed.), Essays in Group Theory, Springer, 1987, pp. 75-265

[10] Gromov, M. Asymptotic invariants of infinite groups (Niblo, G.; Roller, M., eds.), Geometric Group Theory, Cambridge University Press, Cambridge, 1993

[11] M. Gromov, Random walk in random groups, Preprint IHÉS, 2002

[12] Kesten, H. Symmetric random walks on groups, Trans. Amer. Math. Soc., Volume 92 (1959), pp. 336-354

[13] Kesten, H. Full Banach mean values on countable groups, Math. Scand., Volume 7 (1959), pp. 146-156

[14] Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, 2003, ArXiv document | arXiv

[15] Ol'shanskii, A.Yu. Almost every group is hyperbolic, Internat. J. Algebra Comput., Volume 2 (1992) no. 1, pp. 1-17

[16] Papasoglu, P. An algorithm detecting hyperbolicity (Baumslag, G. et al., eds.), Geometric and Computational Perspectives on Infinite Groups, DIMACS Ser. Discrete Math. Theor. Comput. Sci., 25, 1996, pp. 193-200

[17] Short, A. et al. Group Theory from a Geometrical Viewpoint (Ghys, É.; Haefliger, A.; Verjovsky, A., eds.), World Scientific, 1991

[18] Woess, W. Cogrowth of groups and simple random walks, Arch. Math. (Basel), Volume 41 (1983), pp. 363-370

[19] Woess, W. Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Math., 138, Cambridge University Press, 2000

[20] A. Żuk, Property (T) and Kazhdan constants for discrete groups, Preprint, École normale supérieure de Lyon

Cité par Sources :