no. 7
Group Theory
A Wilson group of non-uniformly exponential growth
[Un groupe de Wilson de croissance exponentielle non-uniforme]

Présenté par : Jacques Tits

Bartholdi, Laurent1
1 Department of Mathematics, Evans Hall, U.C. Berkeley, USA
Comptes Rendus. Mathématique, Tome 336 (2003) no. 7, pp. 549-554.

Cette Note construit un groupe W de type fini dont la croissance des boules est exponentielle, mais pour laquelle l'infimum des taux de croissance vaut 1 – en d'autres termes, W est de croissance exponentielle non-uniforme.

Ceci répond à une question de Mikhael Gromov (Structures métriques pour les variétés riemanniennes, in : J. Lafontaine, P. Pansu (Eds.), CEDIC, Paris, 1981).

Cette construction donne aussi un groupe de croissance intermédiaire V ressemblant localement à W dans le sens que (en changeant le système générateur de W) des boules de rayon arbitrairement grand coïncident dans les graphes de Cayley de V et W.

This Note constructs a finitely generated group W whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 – in other words, of non-uniformly exponential growth.

This answers a question by Mikhael Gromov (Structures métriques pour les variétés riemanniennes, in: J. Lafontaine, P. Pansu (Eds.), CEDIC, Paris, 1981).

The construction also yields a group of intermediate growth V that locally resembles W in that (by changing the generating set of W) there are isomorphic balls of arbitrarily large radius in V and W's Cayley graphs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00131-6
Bartholdi, Laurent 1

1 Department of Mathematics, Evans Hall, U.C. Berkeley, USA 
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Bartholdi, Laurent. A Wilson group of non-uniformly exponential growth. Comptes Rendus. Mathématique, Tome 336 (2003) no. 7, pp. 549-554. doi : 10.1016/S1631-073X(03)00131-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00131-6/

[1] L. Bartholdi, Groups of intermediate growth, 2002, submitted

[2] Eskin, A.; Mozes, S.; Oh, H. Uniform exponential growth for linear groups, Internat. Math. Res. Notices, Volume 31 (2002), pp. 1675-1683

[3] Grigorchuk, R.I. On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR, Volume 271 (1983) no. 1, pp. 30-33

[4] Gromov, M. Structures métriques pour les variétés riemanniennes (Lafontaine, J.; Pansu, P., eds.), CEDIC, Paris, 1981

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

[6] Neumann, P.M. Some questions of Edjvet and Pride about infinite groups, Illinois J. Math., Volume 30 (1986) no. 2, pp. 301-316

[7] D.V. Osin, The entropy of solvable groups, Ergodic Theory Dynamical Systems, 2003, to appear

[8] J.S. Wilson, On exponential and uniformly exponential growth for groups, Preprint, 2002, http://www.unige.ch/math/biblio/preprint/2002/growth.ps

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