Let $G$ be a f.g. group and $d$ be a word metric w.r.t. a symmetric generating set. For $g\in G$, define $|g|:=d(g,e)$, where $e$ is the group identity. For $k\in\mathbb N$, put $$n_k:=\#\{g\in G: |g|\leq k\}\quad\text{and}\quad m_k:=\#\{g\in G: |g|=k\}$$ In which groups $\lim_k\frac{m_k}{n_k}=0$?

I suggest deleting the tag decomposition-theorem. I would guess that the tag was intended to be about the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber. It seems unnecessary to have a whole tag just dedicated to this theorem (perverse-sheaves should suffice). What's worse, most po...