Timeline for Right-angled Artin groups that split as direct products
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 4, 2023 at 0:56 | comment | added | Matt Zaremsky | @Mike Maybe Servatius's proof can be simplified in the special case when the element is a product of one copy of each generator, since that's all you care about to get the result you want (i.e., that if the graph is not a join then such an element has cyclic centralizer). You don't need to understand the centralizer of every element. On the other hand, direct products can be deceptively difficult sometimes, so maybe it's just a harder problem than it seems! | |
| Feb 3, 2023 at 18:35 | comment | added | Mike | I'm trying to read Servatius's paper right now, and it's a real bear. Weird notation, very combinatorial, constant "it's easy to see" both for trivial and non-trivial steps, etc. Has no one tried to clean it up? | |
| Feb 3, 2023 at 17:42 | vote | accept | Mike | ||
| Feb 3, 2023 at 15:47 | comment | added | Mike | @AGenevois: I would be very interested in this! | |
| Feb 3, 2023 at 15:46 | comment | added | Mike | Great, thanks! I did a bit of searching, and it's a bit nuts that despite all the work on RAAGs there doesn't seem to be any alternate proof of Servatius's theorem (which is pretty intense and combinatorial) in the literature. | |
| Feb 3, 2023 at 14:31 | comment | added | AGenevois | Yes, an argument based on CAT(0) geometry is possible, but I am not sure it would be easier. I think that an easy and short argument is possible just by using van Kampen diagrams. I will try to write an answer in this direction. | |
| Feb 3, 2023 at 12:28 | comment | added | Matt Zaremsky | @HJRW Good point, right, Servatius's result is the key thing. (I also feel like there should be an easier proof of the original question using CAT(0) stuff, but I guess it would probably also just be "easy" modulo some deep result.) | |
| Feb 3, 2023 at 11:51 | comment | added | HJRW | The proof of Proposition 2.14 (in both Koberda and Behrstock--Charney) is very short modulo "Servatius' centralizer theorem". So it feels like Servatius (23 years before Behrstock--Charney) is really the correct reference. | |
| Feb 3, 2023 at 0:11 | history | answered | Matt Zaremsky | CC BY-SA 4.0 |