Timeline for Examples of locally hyperbolic groups
Current License: CC BY-SA 4.0
12 events
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| Nov 13, 2023 at 18:02 | comment | added | HJRW | @GilesGardam: Yes, this is worth pointing out! Also, I think a recent theorem of Kielak--Linton deals with the case of random few-relator groups with $n=m-1$ that I mentioned in item 6. | |
| Nov 13, 2023 at 16:53 | comment | added | Giles Gardam | For the benefit of future readers: Baumslag's conjecture that one-relator groups are coherent is now a theorem of Jaikin-Zapirain and Linton arxiv.org/abs/2303.05976 | |
| Jul 4, 2022 at 14:45 | history | edited | HJRW | CC BY-SA 4.0 |
Deleted possibly confusing sentence.
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| Jul 4, 2022 at 12:51 | comment | added | HJRW | @Carl-FredrikNybergBrodda: It's unclear if our coherence result is effective. This is also something I would be very interested to know! It certainly doesn't follow immediately from our proof. I think the same is true for Scott for 3-manifolds and for Feighn--Handel for ascending HNN extensions of free groups, but would be interested to hear otherwise. In the 3-manifold context, the presentation problem is known to be solvable (see Remark 4.19 of arxiv.org/abs/1405.6274) but this uses a lot more machinery. | |
| Jul 4, 2022 at 12:45 | comment | added | Carl-Fredrik Nyberg Brodda | Yes, I could cheat by knowing a presentation of $H$ in advance, but I'd rather not :-) Regarding a presentation for $H$, such a finite one exists by Louder-You -- do you know if this result is translatable into something practically effective (for OR groups with torsion)? Some time ago I tried to see if something could be extracted, but the geometric details were (and, I suspect, remain) a bit too much for me. | |
| Jul 4, 2022 at 12:35 | comment | added | HJRW | @Carl-FredrikNybergBrodda: Since you seem to know a presentation for your subgroup $H$ in advance, it can be decided just because a hyperbolic structure for $H$ can be found. But in general, it’s not clear whether or not Gersten’s theorem is effective: given a generating set for a hyperbolic subgroup $H$ of a 2d hyperbolic group, can a presentation for $H$ be computed? Similarly, there’s no known example of a coherent group with unsolvable presentation problem. | |
| Jul 4, 2022 at 12:25 | comment | added | Carl-Fredrik Nyberg Brodda | How "effective" is this decidability? For example, if $G = \langle a, t \mid (t^{-1} a^{-1} t a^2)^2 = 1 \rangle$, and $H$ is the subgroup of $G$ generated by $\{ t, a^2, ata^{-1} \}$, how hard is it to decide (in practice) whether $ata^{-2}t^{-1}a^{-1}t^{-1}$ is conjugate to $a^{-4}t^{-1}$ in $H$? (Example chosen at "random", but spoiler: the answer should be "yes, they are conjugate". However, $H$ is a torsion-free one-relator group, so we cannot rely on Newman's theorem). | |
| Jul 4, 2022 at 9:40 | history | edited | HJRW | CC BY-SA 4.0 |
Added discussion of the membership problem.
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| Jul 4, 2022 at 9:30 | comment | added | HJRW | @Carl-FredrikNybergBrodda: Sure. The 2-free examples (we call them "negative immersions") can also be seen as a much deeper, and much more generic, extension of Newman's theorem. | |
| Jul 4, 2022 at 9:26 | history | edited | HJRW | CC BY-SA 4.0 |
Added reference to Gardam's answer.
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| Jul 4, 2022 at 9:26 | comment | added | Carl-Fredrik Nyberg Brodda | So every finitely generated subgroup of a one-relator group with torsion is hyperbolic! In particular, every such subgroup has decidable conjugacy problem. This is a really nice (recent) extension of Bill Newman’s old result. | |
| Jul 4, 2022 at 9:23 | history | answered | HJRW | CC BY-SA 4.0 |