Group Theory
A Wilson group of non-uniformly exponential growth
Comptes Rendus. Mathématique, Volume 336 (2003) no. 7, pp. 549-554.

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.

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.

Received:
Accepted:
Published online:
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, Volume 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/

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[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

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[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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