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I'm quite a neophyte in this area, so apologies if this question is easy. (I've edited slightly to make my question more clear.)

I've been trying to learn about growth rates for finitely generated groups, and would like to know more about the polynomial regime. Let's fix a group $G$ with finite set $S$ of generators, and denote by $c_n(G)$ the number of group elements representable as a product of less than or equal to $n$ elements of $S$.

It looks like it's known that if $c_n(G) < n^C$ for some constant $C$ and all large $n$, then in fact there exists an integer $d$ (let's call $d$ the 'degree') for which $c_n(G)/n^d$ approaches a constant $K$ (let's call $d$ the 'leading coefficient.')

Question: Is anything known about leading coefficients when $c_n(G)$ grows polynomially? In particular, must they always be at least 1?

I read Bass's original paper on growth rates for nilpotent groups, and if I read correctly, it seems his proof (which might be originally due to Wolf?) shows that nilpotent groups with degree $1$ or $2$ indeed must have leading coefficients at least $1$, and combined with Gromov's theorem, I guess this would show that arbitrary polynomial growth with degree $1$ or $2$ must have leading coefficient at least $1$. But his bounds do not seem to immediately imply this for large degree.

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    $\begingroup$ An infinite locally finite graph always has a geodesic ray from any point. Hence if $G$ is infinite, the growth rate is $\ge n$ w.r.t. any generating subset. $\endgroup$
    – YCor
    Commented Apr 22, 2021 at 16:48
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    $\begingroup$ More generally the degree of polynomial growth is an integer $D$ which is computable (Gromov + Wolf). In addition, Pansu proved that $c_n/n^D$ converges. $\endgroup$
    – YCor
    Commented Apr 22, 2021 at 16:49
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    $\begingroup$ When $G$ is nilpotent and torsion-free, Breuillard-Le Donne characterize the limit as the volume of the unit ball of some space that would make sense if one knew some things that I don't. See the paragraph after Corollary 8.1 on page 15 in "On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry", PNAS 110 (48) 2013 pp. 19220-19226, arxiv:1204.1613. $\endgroup$
    – Sophie M
    Commented Apr 22, 2021 at 19:33
  • $\begingroup$ @Ycor: thanks! I guess my question is about the value this converges to (this is what I called the "leading coefficient.") $\endgroup$
    – Ronnie Pavlov
    Commented Apr 29, 2021 at 20:57
  • $\begingroup$ @Sophie: this is cool, although I also don't know enough to understand exactly what that volume is. $\endgroup$
    – Ronnie Pavlov
    Commented Apr 29, 2021 at 20:58

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