Timeline for What is known about the upper density of torsion elements in finitely generated groups?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 7, 2023 at 20:11 | comment | added | I. Haage | Thanks @YCor, please note: $G$ being infinite is a standing assumption made in the first sentence. | |
| Nov 7, 2023 at 19:35 | comment | added | YCor | 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). | |
| Nov 7, 2023 at 15:08 | comment | added | I. Haage | @YCor: notation enhanced, thank you | |
| Nov 7, 2023 at 15:06 | history | edited | I. Haage | CC BY-SA 4.0 |
incorporated a better notation following YCors's suggestion in the comments
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| Nov 7, 2023 at 13:50 | comment | added | YCor | 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. | |
| Nov 7, 2023 at 13:41 | comment | added | YCor | 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. | |
| Nov 7, 2023 at 10:59 | history | edited | I. Haage | CC BY-SA 4.0 |
added 1 character in body
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| Nov 7, 2023 at 10:50 | history | asked | I. Haage | CC BY-SA 4.0 |