On exponential growth rates for free groups

• Autores: Malik Koubi
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 42, Nº 2, 1998, págs. 499-507
• Idioma: inglés
• DOI: 10.5565/publmat_42298_13
• Títulos paralelos:
  • Tasas de crecimiento exponencial para grupos libres
• Enlaces
  • Texto completo
• Resumen

  • Let $F_p$ be a free group of rank $p \ge 2$. It is well-known that, with respect to a $p$-element generating set, that is, a basis, the exponential growth rate of $F_p$ is $2p-1$. We show that the exponential growth rate $\tau$ 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 $\tau < 2p-1$. We also prove that, for any finite generating set $X$ of $F_p$ which is disjoint from $X^{-1}$, the exponential growth rate $\tau$ of $F_p$ with respect to $X$ is $2p-1$ if and only if $X$ is a basis of $F_p$; otherwise $\tau > 2p-1$.