Timeline for Distortion in the Brin-Thompson 2V
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 15, 2022 at 14:37 | comment | added | Ville Salo | Yes, an element can have no finite orbits, you can simulate the SMART machine from the Turing machine literature | |
| May 15, 2022 at 14:18 | comment | added | Matt Brin | The question about BS(1,2) raises side questions since it asks if an element can be conjugate to its square. There may be easy answers to the following, but if there are, they escape me. Can an element have no finite orbits? If an element has finite orbits, but a bound on the size of its orbits, can it be conjugate to its square? Etc. | |
| May 12, 2022 at 10:58 | comment | added | Matt Zaremsky | @VilleSalo Ah, right, I see. | |
| May 12, 2022 at 10:17 | comment | added | Ville Salo | Also, I haven't read the paper, but as far as I understand this concerns the natural copies of F, T, and V in 2V. I seem to recall that every individual element of V which is of infinite order is a translation on (the germ of) some periodic point, which perhaps implies it cannot be distorted in 2V. | |
| May 12, 2022 at 10:03 | comment | added | Ville Salo | @MattZaremsky: The elements seem to be like $g^n h g^{-n}$ so not powers of a single element. | |
| May 12, 2022 at 10:02 | comment | added | Matt Zaremsky | @AGenevois In that Burillo-Cleary paper you mentioned, doesn't Section 5 show that there are distortion elements? | |
| May 12, 2022 at 7:42 | comment | added | Ville Salo | Sorry I removed my comment asking about connections between CAT(0) cube complexes and distortion, and whether "not acting on any CAT(0) complex" is interesting. (I removed it because after reading your first comment a few times I realized you already answered the technical question.) Thank you for the answer though. | |
| May 12, 2022 at 7:38 | comment | added | AGenevois | Yes, this is a result due to Haglund. I think the question is interesting because: (1) if there exists such a proper action, then this implies interesting properties for 2V automatically; and (2) if there is no such action, then this constrasts with other Thompson-like groups and asks the question of which geometry would be relevant and how it is different from the one-dimensional case. | |
| May 12, 2022 at 7:28 | comment | added | Ville Salo | Thanks for your "no" vote. It's hard to find negative information in papers... I think I did notice the result of Burillo and Cleary, and it's what made me suspect that this is not known. | |
| May 12, 2022 at 7:23 | comment | added | AGenevois | By the way, some elements are proved to be undistorted in Burillo and Cleary's article Metric properties in higher dimensional Thompson's group. | |
| May 12, 2022 at 7:22 | comment | added | AGenevois | Up to my knowledge, this is not known. For many Thompson-like groups, proper actions on CAT(0) cube complexes can be constructed, which implies that infinite-order elements are undistorted. But the usual construction fails in higher dimensions. It is something I would like to know: does 2V acts properly on some CAT(0) cube complex? | |
| May 12, 2022 at 6:54 | history | asked | Ville Salo | CC BY-SA 4.0 |