It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies $(\dagger)\quad |S^n| \le C\cdot n^c$ for all $n\in\mathbb{Z}_{>0}$ and some $C,c\in[0,+\infty[$ then the group actually satisfies that $|S^n| \sim C_S\cdot n^{d_\Gamma}$ for some constants $C_S\in]0,+\infty[$ and $d_\Gamma\in\mathbb{Z}_{\ge 0}$. In particular for any group $\Gamma$ the limit in $[0,+\infty]$ of the sequence $\lim \log|S^n|/\log n$ exists (**), and if finite it is an integer. Is there a known proof of the latter result which does not use Gromov's theorem?
(*) To be self contained: the former states that a group $\Gamma$ satisfying $(\dagger)$ must contain with finite index a nilpotent group, and the latter studies the growth of nilpotent groups and establishes the stated conclusion for such.
(**) To deal with all groups maybe you need an extension of Gromov's result that covers the case where you have the polynomial bound$(\dagger)$ only for an infinite sequence of integers $n$.
The question might appear a bit gratuitous, so I'll say that the motivation for asking this comes from the hope to extend the result about the integral exponent to the generic volume growth of so-called "unimodular random hyperbolic surfaces" (a generalization of infinite covers of compact hyperbolic surfaces).
