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Let $\Gamma$ be a finitely generated group of exponential growth and $gr(S)=\lim_{k\rightarrow \infty} \sqrt[k]{|B_k(S)|}$ be the growth rate of $\Gamma$ with respect to the generating set $S$. I am confused with the following question: Does there always exist a generating set $S'$ such that $$\frac{|B_k(S')|}{gr(S')^k}\rightarrow 1, \text{ when }k\rightarrow \infty$$
Let $\Gamma$ be a finitely generated group of exponential growth and $gr(S)=\lim_{k\rightarrow \infty} \sqrt[k]{|B_k(S)|}$ be the growth rate of $\Gamma$ with respect to the generating set $S$. I am confused with the following question: Does there exist a generating set $S'$ such that $$\frac{|B_k(S')|}{gr(S')^k}\rightarrow 1, \text{ when }k\rightarrow \infty$$