Cogrowth and spectral gap of generic groups
Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 289-317.

The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.

L'exposant de cocroissance d'un groupe contrôle le spectre de la marche aléatoire. Nous prouvons que pour un groupe générique (dans le modèle à densité) cet exposant est arbitrairement proche de celui du groupe libre. En outre, cet exposant est stable par quotient aléatoire d'un groupe hyperbolique sans torsion.

DOI: 10.5802/aif.2099
Classification: 20P05, 20F69, 20F06
Keywords: Random groups, cogrowth, hyperbolic groups, random walk on groups
Mot clés : groupes aléatoires, cocroissance, groupes hyperboliques, marche aléatoire sur les groupes
Ollivier, Yann 1

1 UMPA - CNRS, École normale supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07 (France)
@article{AIF_2005__55_1_289_0,
     author = {Ollivier, Yann},
     title = {Cogrowth and spectral gap of generic groups},
     journal = {Annales de l'Institut Fourier},
     pages = {289--317},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     doi = {10.5802/aif.2099},
     mrnumber = {2141699},
     zbl = {02162474},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2099/}
}
TY  - JOUR
AU  - Ollivier, Yann
TI  - Cogrowth and spectral gap of generic groups
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 289
EP  - 317
VL  - 55
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2099/
DO  - 10.5802/aif.2099
LA  - en
ID  - AIF_2005__55_1_289_0
ER  - 
%0 Journal Article
%A Ollivier, Yann
%T Cogrowth and spectral gap of generic groups
%J Annales de l'Institut Fourier
%D 2005
%P 289-317
%V 55
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2099/
%R 10.5802/aif.2099
%G en
%F AIF_2005__55_1_289_0
Ollivier, Yann. Cogrowth and spectral gap of generic groups. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 289-317. doi : 10.5802/aif.2099. http://archive.numdam.org/articles/10.5802/aif.2099/

[C] J.M. Cohen Cogrowth and Amenability of Discrete Groups, J. Funct. Anal., Volume 48 (1982), pp. 301-309 | DOI | MR | Zbl

[Ch00] C. Champetier L'espace des groupes de type fini, Topology, Volume 39 (2000) no. 4, pp. 657-680 | DOI | MR | Zbl

[Ch93] C. Champetier Cocroissance des groupes à petite simplification, Bull. London Math. Soc., Volume 25 (1993) no. 5, pp. 438-444 | DOI | MR | Zbl

[Ch95] C. Champetier Propriétés statistiques des groupes de présentation finie, J. Adv. Math., Volume 116 (1995) no. 2, pp. 197-262 | MR | Zbl

[GdlH] R.I. Grigorchuk; P. de la Harpe On problems related to growth, entropy, and spectrum in group theory, Dynam. Control Systems, Volume 3 (1997) no. 1, pp. 51-89 | DOI | MR | Zbl

[Gh] É. Ghys Groupes aléatoires (séminaire Bourbaki), Volume 916 (2003)

[Gri] R.I. Grigorchuk; R.L. Dobrushin, Ya.G. Sinai Symmetrical Random Walks on Discrete Groups (Adv. Prob. Related Topics), Volume 6 (1980), pp. 285–325 | Zbl

[Gro03] M. Gromov Random Walk in Random Groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | MR | Zbl

[Gro87] M. Gromov; S.M. Gersten Hyperbolic Groups (Essays in group theory) (1987), pp. 75-265 | Zbl

[Gro93] M. Gromov Asymptotic Invariants of Infinite Groups, Geometric group theory, Cambridge University Press, Cambridge, 1993 | MR

[HLS] N. Higson; V. Lafforgue; G. Skandalis Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 330-354 | DOI | MR | Zbl

[K] H. Kesten Symmetric Random Walks on Groups, Trans. Amer. Math. Soc., Volume 92 (1959), pp. 336-354 | DOI | MR | Zbl

[LS] R.C. Lyndon; P.E. Schupp Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89, 1977 | MR | Zbl

[Oll] Y. Ollivier Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal., Volume 14 (2004) no. 3, pp. 595-679 | MR | Zbl

[Ols] A.Yu. Ol'shanski\u\i Almost Every Group is Hyperbolic, Int. J. Algebra Comput., Volume 2 (1992) no. 1, pp. 1-17 | DOI | MR | Zbl

[P] F. Paulin Sur la théorie élémentaire des groupes libres, Séminaire Bourbaki, Volume 922 (2003)

[Sh] H. Short; et al., Group Theory from a Geometrical Viewpoint, World Scientific, 1991

[W] W. Woess Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, 138, 2000 | MR | Zbl

Cited by Sources: