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Growth rates of amenable groups
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G. N. Arzhantseva
Published/Copyright:
July 27, 2005
Abstract
Let Fm be a free group with m generators and let R be a 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 ρ = ρ(R ) of the shortest non-trivial relation in R. It follows that the growth rate of Fm / R′ approaches 2m – 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 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.
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Published Online: 2005-07-27
Published in Print: 2005-05-25
© de Gruyter
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Articles in the same Issue
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- Actions of relative Weyl groups II
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