I'm quite a neophyte in this area, so apologies if this question is easy.

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 $S$.

If I understand correctly, then 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 this the 'degree') for which $c_n(G)/n^d$ approaches a constant (let's call this the 'leading coefficient.')

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 $1$.

What can be said about possible leading coefficients more generally? Naively, I feel like it's not possible for it to ever be less than $1$, but this is very much not my area.

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.