Let Fp 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 Fp 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 Fp which is disjoint from X-1, the exponential growth rate τ of Fp with respect to X is 2p-1 if and only if X is a basis of Fp; otherwise τ > 2p-1.
@article{urn:eudml:doc:41342,
title = {On exponential growth rates for free groups.},
journal = {Publicacions Matem\`atiques},
volume = {42},
year = {1998},
pages = {499-507},
mrnumber = {MR1676040},
zbl = {0929.20022},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41342}
}
Koubi, Malik. On exponential growth rates for free groups.. Publicacions Matemàtiques, Tome 42 (1998) pp. 499-507. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41342/