Timeline for QI-closure of $\mathrm{NA}\times\mathrm{NA}$
Current License: CC BY-SA 4.0
19 events
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| May 22, 2020 at 10:13 | comment | added | Ville Salo | (I don't really understand your claim about the asymptotic cone, I suggested QI+subgroups cannot give you products due to some Bass-Serre theory arguments I spotted in the literature (and which I don't fully understand either).) | |
| May 22, 2020 at 10:03 | comment | added | Ville Salo | I.e. can distorted subgroups cause a problem | |
| May 22, 2020 at 9:46 | comment | added | Ville Salo | Am I correct that even by repeatedly passing to subgroups and taking QI groups you cannot find such a product? | |
| May 22, 2020 at 8:37 | comment | added | YCor | @VilleSalo a Baumslag-Solitar group $BS(m,n)$ with $|m|\neq |n|$ has a 1-dimensional asymptotic cone. Hence a direct product of two infinite f.g. groups cannot embed QI into it. In particular it cannot be QI to such a direct product. | |
| May 20, 2020 at 11:37 | comment | added | Ville Salo | Ok, I found the understandable classification of those (QI to $\mathrm{BS}(2,3)$ groups). I'll look at this subquestion. The main question was more or less successful: I have not missed anything completely obvious. | |
| May 20, 2020 at 9:39 | comment | added | Ville Salo | @YCor: On second thought, of course you are right, if some group in this class is one for which strongly aperiodic SFTs are not previously known, that's interesting, since indeed my condition implies that. Mainly non-RF Baumslag-Solitar groups spring to mind, are they QI to products of NA? I know they are a single QI-class, but I don't know what else is there. | |
| May 20, 2020 at 3:44 | comment | added | Ville Salo | Regarding last comment of @YCor, I think knowing the exact class doesn't affect the sort of things I'd consider interesting, and does not help suggesting groups in the class. You guess the spirit correctly (though I nitpick that all countable groups admit a SA subshifts, and not all products of f.p. nonamenable groups admit SA SFTs). | |
| May 19, 2020 at 23:35 | comment | added | Ville Salo | @YCor does your class of interesting examples include some where the group is QI to a product of two nonamenables, but you use something other than two trees? | |
| May 19, 2020 at 23:34 | comment | added | Ville Salo | I added the correct BM reference, the deduction is on page 760 of Drutu-Kapovich. | |
| May 19, 2020 at 23:29 | history | edited | Ville Salo | CC BY-SA 4.0 |
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| May 19, 2020 at 22:46 | comment | added | YCor | BTW "interesting" obviously depends on how your class is defined... I somewhat have the guess that your class is that of groups that admit a strongly aperiodic subshift, or something of this spirit, but this is hard to guess a priori... | |
| May 19, 2020 at 20:51 | comment | added | YCor | @LSpice in this case it probably means simple f.p. groups that occur as acting geometrically on a product of two trees. Other interesting groups QI to a product of non-amenable groups (and not virtually directly decomposable) are various irreducible lattices in products of Lie or non-archimedean groups. Rataggi also produced groups QI to a product of two bushy trees, that have no proper finite index subgroups, but have many normal subgroups of infinite index. | |
| May 19, 2020 at 20:50 | comment | added | YCor | Axiom 2 sounds weirdly stated, since $H$ QI to $G$ fp implies $G$ fp. So you have an isomorphism-closed class $\mathcal{G}$ of groups such that its subclass of fp elements is QI-closed, and contains all direct products of 2 fg rp non-amenable groups. The smallest class satisfying thus consists of (a) the fp groups QI to such a fp product (b) the rp fg direct products of two non-amenable groups. Probably the last formulation of the question is better | |
| May 19, 2020 at 20:49 | comment | added | LSpice | There appear to be at least three (1 2 3) "Burger–Mozes"s. Which one is the relevant one here? | |
| May 19, 2020 at 20:48 | history | edited | LSpice | CC BY-SA 4.0 |
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| May 19, 2020 at 20:03 | history | edited | Ville Salo | CC BY-SA 4.0 |
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| May 19, 2020 at 19:54 | history | edited | Ville Salo | CC BY-SA 4.0 |
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| May 19, 2020 at 19:27 | history | edited | Ville Salo | CC BY-SA 4.0 |
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| May 19, 2020 at 19:07 | history | asked | Ville Salo | CC BY-SA 4.0 |