Timeline for Kleinian groups containing an isomorphic copy of itself
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| May 9, 2015 at 7:05 | comment | added | HJRW | @DanielGroves, oh right, of course. | |
| May 8, 2015 at 23:21 | comment | added | Daniel Groves | @HJRW: There are examples on the first page of Delzant and Potyagailo's paper. Alternatively, amalgamate a surface group with Z^2 gluing an element represented by a simple closed curve to a basis element. Now unwrap the unglued direct factor of the Z^2 to get a proper subgroup isomorphic to the whole group. DP's characterization is that this is essentially the only thing that can happen for Kleinian groups. | |
| May 8, 2015 at 19:53 | comment | added | HJRW | @DanielGroves, good point about the torsion. When you say there are freely indecomposable toral relatively hyperbolic groups that aren't co-Hopfian, I assume you're not just talking about the trivial examples $\mathbb{Z}^n$? What's a non-trivial example? | |
| May 8, 2015 at 12:01 | comment | added | Daniel Groves | @HJRW Hi Henry, Actually Sela's work doesn't extend to the toral relatively hyperbolic setting, since there are freely indecomposable toral relatively hyperbolic groups which are not co-Hopfian. Using Sela's shortening argument, however, I believe one could classify the toral relatively hyperbolic groups which are not co-Hopfian in a similar way to Delzant and Potyagailo's characterization (and I guess this is what you meant). Of course, Kleinian groups can have torsion, so are not all toral relatively hyperbolic groups. | |
| May 6, 2015 at 14:13 | comment | added | HJRW | I should probably add that the mistake is documented in Example 1 of the Louder--Touikan paper. | |
| May 6, 2015 at 14:06 | comment | added | Igor Rivin | @HJRW Ah, so! I did NOT know that! | |
| May 6, 2015 at 11:00 | comment | added | HJRW | Another thing to mention is that there may be a problem with the proof of the Delzant--Potyagailo result. It relies on their Topology paper proving 'hierarchical accessibility', which contains an error. This has been fixed by Louder--Touikan (arxiv.org/abs/1302.5451), but one would need to check that the results needed for the Kleinian groups paper remain true. | |
| May 6, 2015 at 10:34 | comment | added | HJRW | It's not true that all Kleinian groups are co-Hopfian -- free groups aren't, for instance. Sela showed that all freely indecomposable hyperbolic groups are co-Hopfian, and his work almost certainly extends to the toral relatively hyperbolic case (which includes all Kleinian groups). | |
| May 6, 2015 at 10:11 | comment | added | Igor Rivin | @YCor I was writing this on no sleep, so simply did not notice Lyonia's name on the paper! The paper of D&P seems to show that Kleinian groups are co-Hopfian, and in the case they are not, the isomorphic copy is NOT of finite index. Presumably, the OP did not ask the question to settle a bar bet, so he does actually want to know the back story. | |
| May 6, 2015 at 10:09 | history | edited | Igor Rivin | CC BY-SA 3.0 |
Fixed attribution
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| May 6, 2015 at 6:39 | comment | added | YCor | @NeilHoffman: what Danny means is that the review says (1st sentence) "A group is co-Hopfian if it is isomorphic to a proper subgroup of itself" while it should be "it is not isomorphic"... | |
| May 6, 2015 at 4:05 | comment | added | Neil Hoffman | @DannyNguyen I don't understand your comment and I would be grateful if you could help me clear up my misunderstandings. First, you have dropped the condition that the subgroup be proper, which seems essential to the definition of co-Hopfian. Second, I don't follow why "is" is in bold in your correction. | |
| May 5, 2015 at 23:07 | comment | added | Danny Nguyen | The MathSciNet review has a typo: "A group is co-Hopfian if it is isomorphic to a subgroup of itself". | |
| May 5, 2015 at 19:54 | comment | added | YCor | This is a yes-or-no question and you provide a link without suggesting what you call "definitive result"... and a quick look at the link does not provide me what you have in mind, except that the link to "Delzant" is a paper by Delzant and Potyagailo (I hate Google for providing only the first name of authors in the list of math papers in Google requests). | |
| May 5, 2015 at 19:50 | vote | accept | Danny Nguyen | ||
| May 5, 2015 at 20:03 | |||||
| May 5, 2015 at 19:35 | history | answered | Igor Rivin | CC BY-SA 3.0 |