# A question about groups of intermediate growth

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?

Update: The answer has been provided by Martin Kassabov below: the condition $\zeta_k(G)=1$ is equivalent for $G$ to have sub-exponential growth.

Valerio

• Nice question. My impression is that there is no simpler proof of 2. You can get some way beyond polynomial growth by quoting Shalom and Tao. Commented Nov 10, 2011 at 11:02
• Commented Nov 10, 2011 at 16:06
• Thank you Andreas, very interesting, even if it seems that doesn't help. Commented Nov 10, 2011 at 17:55

If $G$ has sub exponential growth one has that $\lim_s \sqrt[s]{B(s)}=1$ if you assume that $\zeta_k(G) = c >1$ then by an easy induction we have $B(nk) > K c^n$ which implies that $$\limsup_s \sqrt[s]{B(s)} > \limsup_n \sqrt[nk]{B(nk)} > \limsup_n \sqrt[nk]{kc^n} = \sqrt[k]{c} > 1$$

Therefore $\zeta_k(G) \leq 1$, but it is clear that $\zeta_k(G)\geq 1$, i.e $\zeta_k(G)=1$.

I.e. for any group of sub exponential growth $\zeta_k(G) =1$.

The same is true if you replace the infimum in the the definition of $\zeta_k(G)$ with limsup, but the argument is more involved and uses that sub-multiplicaticity estimates.

• I take the last comment back -- using submultiplicativity one can only show that $lim_k \sqrt[k]{\bar zeta_k(G)} =1$ where $\bar \zeta_k(G) = \limsup_n \frac{B(kn+k)}{B(kn)}$. Commented Nov 10, 2011 at 22:14
• It seems good. This is nice, also because gives a direct proof in the case of polynomial growth, when all people with whom I talked was convinced of the necessity to use both Gromov and Pansu theorem. Commented Nov 10, 2011 at 22:45
• When making my comment above I was addressing the question for "supremum" instead of "infimum". I feel one does need Gromov/Pansu to do that. Commented Nov 11, 2011 at 10:02
• @martin: do you think one use your corollary 1.3 to exibit at least a group of intermediate growth such that $\overline\zeta(G)=1$? Commented Nov 11, 2011 at 19:50
• @valerio: I think most (if not all) groups of intermediate growth satisfy $\bar \zeta_k=1$. Our result with Igor is not strong enough to get the value of $\bar \zeta$ but my guess is that it is also 1. I feel that there are no groups (of intermediate growth) with $\bar \zeta >1$. Commented Nov 11, 2011 at 23:41

Martin Kassabov just gave a seminar this Tuesday in Royal Holloway about joint work with Igor Pak, see http://arxiv.org/PS_cache/arxiv/pdf/1108/1108.0262v1.pdf. I think it might be useful to answer your question, but you will have to check the details.

• it seems to be very useful. In particular, corollary 1.3. Tomorrow I will have a look at the details. Thank you very much. Commented Nov 10, 2011 at 18:14
• Yiftach: Thanks for advertising my work, but I think it has nothing to do with question Commented Nov 10, 2011 at 22:11
• As long as I spelled your name correctly... Anyhow, your results, even if do not answer this questions directly, are interesting and relevant to the topic. Commented Nov 11, 2011 at 11:58