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Let $G$ be a finitely generated group of polynomial growth. This means that the size $B_n$ of the ball of radius $n$ satsifies: $$ A n^d \leq B_n \leq Bn^d $$ for some constants $A$, $B$.

My question is, what can be said about the ratio $A/B$? Is there a uniform bound (independent of the group) depending maybe on the nilpotent structure of the group or some other parameters, but not otherwise on the group (this would mean that the size of balls cannot oscillate too wildly)?

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I think Breuillard proved that $B_n/n^d$ converges. –  YCor May 11 at 21:42
    
@YvesCornulier: any reference for that? –  Michal Kotowski May 11 at 22:00
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