Arzhantseva, G. N.; Guba, V. S.; Guyot, L. Growth rates of ameable groups. (English) Zbl 1078.20030 J. Group Theory 8, No. 3, 389-394 (2005). Summary: Let \(F_m\) be a free group with \(m\) generators and let \(R\) be a normal subgroup such that \(F_m/R\) projects onto \(\mathbb{Z}\). 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. Cited in 4 Documents MSC: 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory 43A07 Means on groups, semigroups, etc.; amenable groups 20F14 Derived series, central series, and generalizations for groups 20F19 Generalizations of solvable and nilpotent groups Keywords:growth rates; derived series; lengths of relations; amenable groups; Abelian-by-nilpotent groups; virtually metabelian groups PDFBibTeX XMLCite \textit{G. N. Arzhantseva} et al., J. Group Theory 8, No. 3, 389--394 (2005; Zbl 1078.20030) Full Text: DOI arXiv