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.