Let $G$ be a countable group that is finitely generated and let $S = \{s_1, \dots, s_d\}$ be a generating set. Suppose also that $S$ is closed under inverses.
Consider now $\Gamma(G,S) = (V, E)$ the Cayley graph associated with $G$ and $S$, where $V = G$ and $E = \{(g, gs)\} \,:\, g\in G, s \in S\}$. Since we are assuming that the generating set $S$ is symmetric, $\Gamma(G, S)$ is an undirected graph.
Now, given $k \geq 0$, let $B_k$ be the ball centered at the identity element of $G$ and radius $k$ on the Cayley graph accordingly to the word metric.
My question is: under what conditions on the growth rate of the group can we guarantee that given a fixed $k_0 \in \mathbb{N}$, $\lim_{k \to +\infty} \frac{|B_{k - k_0}|}{|B_k|} = 1$?
I know, for exemple, that if $|B_n| = n^{{a}}$, for some $a$, this would hold. How much can we generalize it?
Any help is appreciated!