When talking about the growth of a group, the cases of exponential and polynomial growth are described by concrete bounds on the growth function, however, this is not the case for intermediate growth (or at least, not that I know), so my question is the following:
Let $G$ be finitely generated group of subexponential growth, meaning that there is a symmetric, finite, generating set $S$ containing the identity and such that $\lim |S^n|^{1/n}=1$.
Does this implies that we can bound the growth of $S$ by $|S^n|\leq C\exp(bn^a)$, for appropiate constants $C>0,b>0,0<a<1$? If not, what about other types of bounds?