Let $T\subset G$ be the set of all torsion elements in a finitely generated infinite group $G$, and let $B_n\subset G$ be the closed ball of radius $n$ around $1$ w.r.t. to the word metric for some choice of a finite generating set $S$. Consider the upper density
$$
t_S(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}.
$$
In the following basic cases $t=t_S$ is independent of $S$:

 - $t(G)=0$ if $G$ is torsion-free (trivially)
 - $t(G)=1$ if $G$ is a [finitely generated infinite torsion group][1] (trivially)
 - $t(G\ast H)=0$ whenever $t(G)=0$ and $H$ is finite (slighly less obvious, can be shown using [Kurosh's Theorem][2])

> **Questions**
> 
>  1. Is $t_S(G)$ always independent of the choice of a generating set $S$?
>  2. Is the $\limsup$ always a proper limit?
>  3. What is known about the values $t_S\in[0,1]$ that occur?

  [1]: https://en.wikipedia.org/wiki/Burnside_problem
  [2]: https://en.wikipedia.org/wiki/Kurosh_subgroup_theorem