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incorporated a better notation following YCors's suggestion in the comments
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I. Haage
I. Haage
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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 $$ t(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$$$ t_S(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$ The extreme values $t=0$ and $t=1$ are attained trivially for (torsion-)free groups and finitely generated infinite torsion groups respectively. Slightly less obvious is that $t(G\ast H)=0$ whenever $t(G)=0$ andIn the following basic cases $H$$t=t_S$ is finite, which one can show usingindependent of Kurosh's Theorem.$S$:

Questions

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

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. Consider the upper density $$ t(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$ The extreme values $t=0$ and $t=1$ are attained trivially for (torsion-)free groups and finitely generated infinite torsion groups respectively. Slightly less obvious is that $t(G\ast H)=0$ whenever $t(G)=0$ and $H$ is finite, which one can show using Kurosh's Theorem.

Questions

  1. Is the value of $t$ independent of the choice of a generating set for $G$?
  2. Is the $\limsup$ always a proper limit?
  3. What is known about the values $t\in[0,1]$ that occur?

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?
added 1 character in body
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I. Haage
I. Haage
  • 233
  • 3
  • 6

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. Consider the upper density $$ t(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$ The extreme values $t=0$ and $t=1$ are attained trivially for (torsion-)free groups and finitely generated infinite torsion groups respectively. Slightly less obvious is that $t(G\ast H)=0$ whenever $t(G)=0$ and $H$ is finite, which one can show using Kurosh's Theorem.

Questions

  1. Is the value of $t$ idependentindependent of the choice of a generating set for $G$?
  2. Is the $\limsup$ always a proper limit?
  3. What is known about the values $t\in[0,1]$ that occur?

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. Consider the upper density $$ t(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$ The extreme values $t=0$ and $t=1$ are attained trivially for (torsion-)free groups and finitely generated infinite torsion groups respectively. Slightly less obvious is that $t(G\ast H)=0$ whenever $t(G)=0$ and $H$ is finite, which one can show using Kurosh's Theorem.

Questions

  1. Is the value of $t$ idependent of the choice of a generating set for $G$?
  2. Is the $\limsup$ always a proper limit?
  3. What is known about the values $t\in[0,1]$ that occur?

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. Consider the upper density $$ t(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$ The extreme values $t=0$ and $t=1$ are attained trivially for (torsion-)free groups and finitely generated infinite torsion groups respectively. Slightly less obvious is that $t(G\ast H)=0$ whenever $t(G)=0$ and $H$ is finite, which one can show using Kurosh's Theorem.

Questions

  1. Is the value of $t$ independent of the choice of a generating set for $G$?
  2. Is the $\limsup$ always a proper limit?
  3. What is known about the values $t\in[0,1]$ that occur?
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I. Haage
I. Haage
  • 233
  • 3
  • 6

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. Consider the upper density $$ t(G)=\limsup_{n\rightarrow\infty}\frac{|B_n\cap T|}{|B_n|}. $$ The extreme values $t=0$ and $t=1$ are attained trivially for (torsion-)free groups and finitely generated infinite torsion groups respectively. Slightly less obvious is that $t(G\ast H)=0$ whenever $t(G)=0$ and $H$ is finite, which one can show using Kurosh's Theorem.

Questions

  1. Is the value of $t$ idependent of the choice of a generating set for $G$?
  2. Is the $\limsup$ always a proper limit?
  3. What is known about the values $t\in[0,1]$ that occur?