On exponential growth rates for free groups

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 $\tau > 2p-1$.