[Un groupe de Wilson de croissance exponentielle non-uniforme]
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.
Accepté le :
Publié le :
Laurent Bartholdi 1
@article{CRMATH_2003__336_7_549_0,
author = {Laurent Bartholdi},
title = {A {Wilson} group of non-uniformly exponential growth},
journal = {Comptes Rendus. Math\'ematique},
pages = {549--554},
publisher = {Elsevier},
volume = {336},
number = {7},
year = {2003},
doi = {10.1016/S1631-073X(03)00131-6},
language = {en},
}
Laurent Bartholdi. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00131-6/
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