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incorporated a better notation following YCors's suggestion in the comments
I. Haage
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What is known about the upper density of torsion elements in finitely generated groups?

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 $S$:

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?
I. Haage
  • 233
  • 3
  • 6