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ON EXPONENTIAL GROWTH RATES FOR FREE GROUPS
Vol. 42, No. 2 (1998), pp. 499-507
Published by: Universitat Autònoma de Barcelona
Stable URL: http://www.jstor.org/stable/43736643
Page Count: 9
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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.
Publicacions Matemàtiques © 1998 Universitat Autònoma de Barcelona