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Title 
Growth rates of amenable groups 

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Published in  Journal of Group Theory. 2005, vol. 8, no. 3, p. 389394  
Abstract  Let $F_m$ be a free group with $m$ generators and let $R$ be its normal subgroup such that $F_m/R$ projects onto $zz$. 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 $ ho= ho(R)$ of the shortest nontrivial relation in $R$. It follows that the growth rate of $F_m/R'$ approaches $2m1$ as $ ho$ approaches infinity. This implies that the growth rate of an $m$generated amenable group can be arbitrarily close to the maximum value $2m1$. This answers an open question by P. de la Harpe. In fact we prove that such groups can be found already in the class of abelianbynilpotent groups as well as in the class of finite extensions of metabelian groups.  
Stable URL  http://archiveouverte.unige.ch/unige:12222  
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arXiv: math/0406013v3 

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