Timeline for Subgroups of torsion-free hyperbolic groups versus subgroups of hyperbolic groups
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2019 at 19:04 | comment | added | user6976 | The answer is yes for small cancelation groups (Gersten 96): every f.p. subgroup of a small cancelation group is itself hyperbolic. | |
| Nov 8, 2019 at 16:14 | comment | added | YCor | I insist but it would be nice to settle the virtually torsion-free case. Indeed, there exist a hyperbolic (indeed virtually free) group $G$ with a torsion-free (normal) subgroup $N$ not contained in any torsion-free finite index subgroup of $G$. Namely, choose a $k$-generated infinite simple group with an element of order 2 (there are many); surject $G=F_k\ast C_2$ onto it with $C_2$ mapped injectively, and let $N$ be the kernel. Then $N$ is torsion-free and the only finite index subgroup of $G$ containing $N$ is $G$ itself, which is not torsion-free. | |
| Nov 8, 2019 at 14:05 | comment | added | ADL | @YCor Thanks. Yes, the other questions make sense and I would be interested to know their answer also. I only asked one question as I didn't want to make the question too convoluted (and this specific question cropped up in my work). | |
| Nov 8, 2019 at 13:57 | comment | added | YCor | Nice question! it also makes sense for finitely generated subgroups, and even arbitrary subgroups. I'm not even sure to be able to answer in the probably easier case of a torsion-free subgroup of a virtually torsion-free hyperbolic groups. At least if this case is doable, then no counterexample to your example is known, since it's unknown whether every hyperbolic f.g. group is virtually torsion-free. | |
| Nov 8, 2019 at 13:54 | history | edited | YCor | CC BY-SA 4.0 |
romanized the abbreviation "tf"
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| Nov 8, 2019 at 13:49 | history | asked | ADL | CC BY-SA 4.0 |