A finitary version of Gromov's polynomial growth theorem

We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d>0$: If there is some $R_0>\exp (\exp (C{d}^C))$ for which the number of elements in a ball of radius $R_0$ in a Cayley graph of $G$ is bounded by $R_0^d$, then $G$ has a finite index subgroup which is nilpotent (of step $<C^d$). An effective bound on the finite index is provided if “nilpotent” is replaced by 'polycyclic", thus yielding a non-trivial result for finite groups as well.