 
Summary: GROWTH RATES OF AMENABLE GROUPS
G. N. ARZHANTSEVA, V. S. GUBA, AND L. GUYOT
Abstract. Let Fm be a free group with m generators and let R
be its normal subgroup such that Fm/R projects onto Z. We give
a lower bound for the growth rate of the group Fm/R # (where R # is
the derived subgroup of R) in terms of the length # = #(R) of the
shortest nontrivial relation in R. It follows that the growth rate
of Fm/R # approaches 2m1 as # approaches infinity. This implies
that the growth rate of an mgenerated 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 abelianbynilpotent groups as well as in
the class of virtually metabelian groups.
1. Introduction
Let G be a finitely generated group and A a fixed finite set of gener
ators for G. We denote by #(g) the word length of an element g # G in
the generators A, i.e. the length of a shortest word in the alphabet A ±1
representing g. Let B(n) denote the ball {g # G  #(g) # n} of radius n
in G with respect to A. The growth rate of the pair (G, A) is the limit
#(G, A) = lim
