This question arose in the comments of A question about groups of intermediate growth. I think it might be interesting to put it more in evidence.

Let $G$ be a f.g. group with a fixed symmetric set of generators $S$ and denote by $B(n)$ the ball of radius $n$ about the identity w.r.t. the word metric induced by $S$.

Fix an integer $k\geq1$ and define $\overline\zeta_k(G)=\lim\sup_n\frac{|B(nk+k)|}{|B(nk)|}$.

Observe that

- If $G$ has polynomial growth, then $\overline\zeta_k(G)=1$, for all $k$.
- If $\overline\zeta_k(G)=1$ for all $k$, then $G$ has sub-exponential growth.

General question:What can we say about $\overline\zeta_k(G)$ if $G$ has intermediate growth?

Martin Kassabov, in the comment to my question, suspects that it should be always (or most of the times) equals to $1$, but I cannot find even a single examples of a group of intermediate growth for which it is equal to $1$. I have to say that my knowledge about groups of intermediate growth is very little and I just tried to apply Corollary 1.3 in http://arxiv.org/PS_cache/arxiv/pdf/1108/1108.0262v1.pdf, but, as already observed by Martin, it is not strong enough to give an example of groups of intermediate growth whose $\overline\zeta_k(G)=1$.

Particular question:Is there an example of group of intermediate growth for which $\overline\zeta_k(X)=1$, for all $k$?

Thanks in advance,

Valerio