Growth rates of amenable groups

Let F-m be a free group with m generators and let R be a normal subgroup such that F-m/R projects onto Z. We give a lower bound for the growth rate of the group F-m/R' (where R' is the derived subgroup of R) in terms of the length rho = rho(R) of the shortest non-trivial relation in R. It follows that the growth rate of F-m/R' approaches 2m - 1 as rho approaches infinity. This implies that the growth rate of an m-generated amenable group can be arbitrarily close to the maximum value 2m - 1. This answers an open question of P. de la Harpe. We prove that such groups can be found in the class of abelian-by-nilpotent groups as well as in the class of virtually metabelian groups.