G. N. Arzhantseva, V. S. Guba, L. Guyot
July 27, 2005

### Abstract

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 ρ = ρ ( R ) of the shortest non-trivial relation in R . It follows that the growth rate of F m / R′ approaches 2 m – 1 as ρ approaches infinity. This implies that the growth rate of an m -generated amenable group can be arbitrarily close to the maximum value 2 m – 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.