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 2m-1 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.
Section de Math’ ematiques, Universit’ e de Gen‘ eve, CP 64, 1211 Gen‘ eve
4, Switzerland
E­mail address: Goulnara.Arjantseva@math.unige.ch
Department of mathematics, Vologda State University, 6 S. Orlov
St., Vologda, 160600, Russia
E­mail address: guba@uni­vologda.ac.ru
Section de Math’ ematiques, Universit’ e de Gen‘ eve, CP 64, 1211 Gen‘ eve

Source: Arzhantseva, Goulnara N. - Section de Mathématiques, Université de Genève

Collections: Mathematics