Timeline for Existence of finite index torsion-free subgroups of hyperbolic groups
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 1, 2022 at 15:31 | history | edited | YCor | CC BY-SA 4.0 |
fixed wording
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| Sep 1, 2022 at 6:49 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question has been bumped anyway)
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| Apr 4, 2011 at 23:39 | comment | added | Ian Agol | Wise has announced results proving residual finiteness for large classes of hyperbolic groups (ones with a quasi-convex hierarchy). In particular, his groups act properly cocompactly on a CAT(0) cube complex. Given his work, I think it is a reasonable conjecture that hyperbolic groups which are the fundamental group of a CAT(0) cube complex are residually finite. I think most people expect general hyperbolic groups to not be residually finite though. ams.org/mathscinet-getitem?mr=2558631 | |
| Sep 27, 2010 at 17:06 | comment | added | Igor Belegradek | To add on Henry's comment: Wise constructed a CAT(0) group with no finite quotients. Another example of such groups are due to Burger-Moses; their groups are lattices in products of trees, and are simple, hence they have no proper finite index subgroups. See Bridson's paper arxiv.org/pdf/math/9810188 and references therein. | |
| Sep 27, 2010 at 15:58 | vote | accept | Dmitri Panov | ||
| Sep 27, 2010 at 14:48 | comment | added | HJRW | On the CAT(0) side, let me just note that Wise constructed a fp CAT(0) group that isn't residually finite. This seems like weak evidence that, in the non-positively curved setting, your question might have a negative answer. | |
| Sep 27, 2010 at 14:18 | answer | added | user6976 | timeline score: 21 | |
| Sep 27, 2010 at 11:53 | comment | added | Sam Nead | That should be "Thanks to Gromov". :) | |
| Sep 27, 2010 at 11:52 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
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| Sep 27, 2010 at 11:39 | answer | added | Sam Nead | timeline score: 8 | |
| Sep 27, 2010 at 11:34 | comment | added | Sam Nead | I don't think that Gromov's example is hyperbolic. Infinite hyperbolic groups contain elements of infinite order. | |
| Sep 27, 2010 at 11:14 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
added 187 characters in body
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| Sep 27, 2010 at 11:05 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
added 170 characters in body
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| Sep 27, 2010 at 11:03 | comment | added | Dmitri Panov | Michele, merci! This is a good reference. Though I really want some positive statements :). So I'll add a condition that the group is finitely presented. | |
| Sep 27, 2010 at 10:47 | comment | added | user47274 | Gromov gave an example of a f.g. infinite torsion group acting on a space of nonpositive curvature in §4.5C of Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75--263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987 | |
| Sep 27, 2010 at 9:54 | history | asked | Dmitri Panov | CC BY-SA 2.5 |