Let $G$ be a finitely generated group with polynomial growth rate. Denote the growth rate by $\Theta(n^k)$ for some specific set of generators.
Consider the sizes of spheres of a given radius $n$ (rather than considering balls as in the definition of growth rate). Is it then true that the size of a sphere is $O(n^{k-1})$?
It seems plausible, and this is what happens with free abelian groups, but theoretically it may be possible that some sphere of one specific radius is very large.