Cogrowth and spectral gap of generic groups

Yann Ollivier[1]

  • [1] UMPA - CNRS, École normale supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07 (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 1, page 289-317
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Ollivier, Yann. "Cogrowth and spectral gap of generic groups." Annales de l’institut Fourier 55.1 (2005): 289-317. <http://eudml.org/doc/116189>.

@article{Ollivier2005,
abstract = {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.},
affiliation = {UMPA - CNRS, École normale supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07 (France)},
author = {Ollivier, Yann},
journal = {Annales de l’institut Fourier},
keywords = {Random groups; cogrowth; hyperbolic groups; random walk on groups; random groups; random walks on groups; cogrowth exponents; free groups},
language = {eng},
number = {1},
pages = {289-317},
publisher = {Association des Annales de l'Institut Fourier},
title = {Cogrowth and spectral gap of generic groups},
url = {http://eudml.org/doc/116189},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Ollivier, Yann
TI - Cogrowth and spectral gap of generic groups
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 1
SP - 289
EP - 317
AB - 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.
LA - eng
KW - Random groups; cogrowth; hyperbolic groups; random walk on groups; random groups; random walks on groups; cogrowth exponents; free groups
UR - http://eudml.org/doc/116189
ER -

References

top
  1. J.M. Cohen, Cogrowth and Amenability of Discrete Groups, J. Funct. Anal. 48 (1982), 301-309 Zbl0499.20023MR678175
  2. C. Champetier, Cocroissance des groupes à petite simplification, Bull. London Math. Soc. 25 (1993), 438-444 Zbl0829.20046MR1233406
  3. C. Champetier, Propriétés statistiques des groupes de présentation finie, J. Adv. Math. 116 (1995), 197-262 Zbl0847.20030MR1363765
  4. C. Champetier, L'espace des groupes de type fini, Topology 39 (2000), 657-680 Zbl0959.20041MR1760424
  5. R.I. Grigorchuk, P. de la Harpe, On problems related to growth, entropy, and spectrum in group theory, Dynam. Control Systems 3 (1997), 51-89 Zbl0949.20033MR1436550
  6. É. Ghys, Groupes aléatoires, 916 (2003) 
  7. R.I. Grigorchuk, Symmetrical Random Walks on Discrete Groups, 6 (1980), Dekker Zbl0475.60007
  8. M. Gromov, Hyperbolic Groups, (1987), 75-265, Springer Zbl0634.20015
  9. M. Gromov, Asymptotic Invariants of Infinite Groups, (1993), Cambridge University Press, Cambridge Zbl0841.20039MR1253544
  10. M. Gromov, Random Walk in Random Groups, Geom. Funct. Anal. 13 (2003), 73-146 Zbl1122.20021MR1978492
  11. N. Higson, V. Lafforgue, G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), 330-354 Zbl1014.46043MR1911663
  12. H. Kesten, Symmetric Random Walks on Groups, Trans. Amer. Math. Soc. 92 (1959), 336-354 Zbl0092.33503MR109367
  13. R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, 89 (1977) Zbl0368.20023MR577064
  14. Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, Geom. Funct. Anal. 14 (2004), 595-679 Zbl1064.20045MR2100673
  15. A.Yu. Ol’shanskiĭ, Almost Every Group is Hyperbolic, Int. J. Algebra Comput. 2 (1992), 1-17 Zbl0779.20016MR1167524
  16. F. Paulin, Sur la théorie élémentaire des groupes libres, Séminaire Bourbaki 922 (2003) 
  17. H. Short, et al., (1991), World Scientific 
  18. W. Woess, Random Walks on Infinite Graphs and Groups, 138 (2000) Zbl0951.60002MR1743100

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.