Timeline for Nontriviality of one-relator products
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 29, 2022 at 5:39 | history | edited | Shijie Gu |
Add tag geometric topology
|
|
| Aug 25, 2022 at 4:39 | vote | accept | Shijie Gu | ||
| Aug 24, 2022 at 20:40 | comment | added | Ian Agol | Corollary V.9.4 of Lyndon-Schupp implies that for a C’(1/6) free product of groups, each factor embeds and hence the group is non-trivial. link.springer.com/book/10.1007/978-3-642-61896-3 | |
| Aug 24, 2022 at 20:29 | comment | added | Anton Klyachko | I do not know. The question is a weaker form of Levin’s conjecture: if $A$ and $B$ are torsion-free groups and $w$ is not conjugate to elements of $A$, then $A$ embeds into the quotient. | |
| Aug 23, 2022 at 18:35 | comment | added | HJRW | Well Klyachko is sometimes active on MO, so perhaps he can comment! | |
| Aug 23, 2022 at 15:10 | comment | added | ADL | @HJRW I can't see this immediately, so I'd have to think properly about it. Theorem 6.1 of Fenn and Rourke's paper proves the result for words $w$ of a certain form, but really, both in this theorem and in their whole paper, they're focusing on embedding $A$ (i.e. proving a Freiheitssatz) rather than proving non-triviality. So their methods are stronger than needed here, which makes me less confident that they can be applied. | |
| Aug 23, 2022 at 13:23 | comment | added | HJRW | @ADL: For some reason I had it in mind that Klyahcko's solution should also work for "one-relator free products" as well as "one-relator HNN extensions", especially with the torsion hypothesis. :) Anyway, your answer certainly covers most situations. | |
| Aug 23, 2022 at 9:19 | comment | added | ADL | @HJRW It isn't a complete solution, so wasn't sure! I've posted it now. | |
| Aug 23, 2022 at 9:17 | answer | added | ADL | timeline score: 6 | |
| Aug 23, 2022 at 2:26 | comment | added | HJRW | @ADL: since this answers the question, you should post it as an answer! | |
| Aug 22, 2022 at 9:11 | history | edited | YCor | CC BY-SA 4.0 |
formatting
|
| Aug 22, 2022 at 8:33 | comment | added | ADL | There are no counter-examples for $A$ torsion-free and $B\cong\mathbb{Z}$. This corresponds to the Kervaire-Laudenbach Conjecture for torsion-free groups, which was proven by Klyachko; see Fenn and Rourke, Klyachko's methods and the solution of equations over torsion-free groups, Enseign. Math. (2) 42 (1996), no. 1-2, 49–74. | |
| Aug 22, 2022 at 6:31 | history | edited | Shijie Gu | CC BY-SA 4.0 |
added 2 characters in body
|
| Aug 22, 2022 at 3:27 | history | asked | Shijie Gu | CC BY-SA 4.0 |