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On montre qu’un groupe hyperbolique non élémentaire est à croissance uniformément exponentielle, c’est-à-dire qu’il existe une constante strictement plus grande que 1, ne dépendant que du groupe , telle que le taux de croissance exponentiel de relatif à n’importe quel système générateur est plus grand que . On redémontre ce faisant qu’un groupe hyperbolique n’a qu’un nombre fini de classes de conjugaison de sous-groupes finis.
Let F be a free group of rank p ≥ 2. It is well-known that, with respect to a p-element generating set, that is, a basis, the exponential growth rate of F is 2p-1. We show that the exponential growth rate τ of a group G with respect to a p-element generating set X is 2p-1 if and only if G is free on X; otherwise τ < 2p-1. We also prove that, for any finite generating set X of F which is disjoint from X, the exponential growth rate τ of F with respect to X is 2p-1 if and only if X is a basis...
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