  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(Cd^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.
