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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?
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  • $\begingroup$ The notation is confusing: you should denote it $t_S(G)$ for $S=B_1$. Then Question 1 is whether $t_S(G)$ only depends on $G$. Question 3 asks what values are obtained when $G$ and $S$ vary. $\endgroup$
    – YCor
    Commented Nov 7, 2023 at 13:41
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    $\begingroup$ A 2007 paper by P. Dani (J. Algebra, arXiv) establishes that for virtually nilpotent groups, $t_S(G)$ only depends on $G$ (hence can be denoted $t(G)$ in this special case), is a genuine limit, and, when $G$ varies, achieves precisely all rational numbers in $[0,1]$ ($1$ if and only if the group is finite) — and more precisely all are achieved by some virtually abelian group. $\endgroup$
    – YCor
    Commented Nov 7, 2023 at 13:50
  • $\begingroup$ @YCor: notation enhanced, thank you $\endgroup$
    – I. Haage
    Commented Nov 7, 2023 at 15:08
  • $\begingroup$ Just to be precise about the three examples items. 1st item: $t(G)=0$ if $G$ is torsion free and infinite (i.e. nontrivial). 2nd item: for $G$ torsion, $t(G)=1$ (no need to assume $G$ infinite). $\endgroup$
    – YCor
    Commented Nov 7, 2023 at 19:35
  • $\begingroup$ Thanks @YCor, please note: $G$ being infinite is a standing assumption made in the first sentence. $\endgroup$
    – I. Haage
    Commented Nov 7, 2023 at 20:11

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