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.
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
@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}, zbl = {02162474}, mrnumber = {2141699}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.2099/
[C] Cogrowth and Amenability of Discrete Groups, J. Funct. Anal., Volume 48 (1982), pp. 301-309 | DOI | MR | Zbl
[Ch00] L'espace des groupes de type fini, Topology, Volume 39 (2000) no. 4, pp. 657-680 | DOI | MR | Zbl
[Ch93] Cocroissance des groupes à petite simplification, Bull. London Math. Soc., Volume 25 (1993) no. 5, pp. 438-444 | DOI | MR | Zbl
[Ch95] Propriétés statistiques des groupes de présentation finie, J. Adv. Math., Volume 116 (1995) no. 2, pp. 197-262 | MR | Zbl
[GdlH] 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] Groupes aléatoires (séminaire Bourbaki), Volume 916 (2003)
[Gri] Symmetrical Random Walks on Discrete Groups (Adv. Prob. Related Topics), Volume 6 (1980), pp. 285325 | Zbl
[Gro03] Random Walk in Random Groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | MR | Zbl
[Gro87] Hyperbolic Groups (Essays in group theory) (1987), pp. 75-265 | Zbl
[Gro93] Asymptotic Invariants of Infinite Groups, Geometric group theory, Cambridge University Press, Cambridge, 1993 | MR
[HLS] Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 330-354 | DOI | MR | Zbl
[K] Symmetric Random Walks on Groups, Trans. Amer. Math. Soc., Volume 92 (1959), pp. 336-354 | DOI | MR | Zbl
[LS] Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89, 1977 | MR | Zbl
[Oll] Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal., Volume 14 (2004) no. 3, pp. 595-679 | MR | Zbl
[Ols] Almost Every Group is Hyperbolic, Int. J. Algebra Comput., Volume 2 (1992) no. 1, pp. 1-17 | DOI | MR | Zbl
[P] Sur la théorie élémentaire des groupes libres, Séminaire Bourbaki, Volume 922 (2003)
[Sh] , Group Theory from a Geometrical Viewpoint, World Scientific, 1991
[W] Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, 138, 2000 | MR | Zbl
Cited by Sources: