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Growth rates of ameable groups. (English) Zbl 1078.20030

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.

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
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