Growth rates of amenable groups

Abstract

Let $F_m$ be a free group with $m$ generators and let $R$ be its normal subgroup such that $F_m/R$ projects onto $\zz$. 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 nontrivial 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 by P. de la Harpe. In fact we prove that such groups can be found already in the class of abelian-by-nilpotent groups as well as in the class of finite extensions of metabelian groups.