On Degrees of Growth of Finitely Generated Groups

We prove that for an arbitrary function $\rho$ of subexponential growth there exists a group $G$ of intermediate growth whose growth function satisfies the inequality $v_{G,S}(n)\ge\rho(n)$ for all $n$. For every prime $p$, one can take $G$ to be a $p$-group; one can also take a torsion-free group $G$. We also discuss some generalizations of this assertion.