Let $G$ be a finitely generated group, $S$ a fixed symmetric generating set and $B(n)$ the ball of radius $n$ about the identity with respect to the word length induced by $S$ on $G$.

Fix $k\geq1$ and denote by $\zeta_k(G,d_S)$ the infimum over $n\geq1$ of $\frac{|B(nk+k)|}{|B(nk)|}$.

Observe that:

1. $\zeta_k(G,d_S)=1$ for all $k$ (well, $k=1$ is enough) implies that $G$ has sub-exponential growth.
2. If $G$ has polynomial growth, then $\zeta_k(G,d_S)=1$, for all $k$ (Gromov + Pansu - by the way, is there a direct proof of this fact, without using such a big theorems?).

What happens in the middle? More formally:

Question: What can we say about $\zeta_k(G,d_S)$ when $G$ has intermediate growth? Is it always $1$? Is it always $>1$? Can be both?

I've asked a couple of persons and it seems that the question may be open, or, at least, non trivial. I don't have a precise idea, since I don't know basically anything about groups of intermediate growth. Does anybody know something? Reference for similar problems?

Thanks in advance,

Valerio