Timeline for Which properties can be read off the balls of a Cayley graph?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 15, 2022 at 20:30 | vote | accept | ARG | ||
| Feb 15, 2022 at 14:07 | answer | added | HJRW | timeline score: 3 | |
| Feb 13, 2022 at 20:53 | comment | added | ARG | @HJRW but if I understand you correctly, there are indeed only finitely many groups with at most $k$ generators, and at most $n$ relations of length at most $\ell$, so the question is then trivial. So actually one of the three should not be a given (and it's probably $n$) | |
| Feb 13, 2022 at 20:49 | comment | added | ARG | @HJRW well, I should have added the open-question tag then, because I definitively did not mean this to be a very precise question. I emphasised this too in the OP [e.g. with labeled/unlaneled]. Also I might very well have made a mistake by saying hyperbolicity fits the bill; this is why I explicitly wrote "I believe[!]" and not "I have proved" or "obviously". Actually another mathematician told me this is so, but he did not give any details (so maybe I misunderstood him). As for the paper of Ozawa, I'm not sure how to check the "sum of squares" inside a finite ball. | |
| Feb 13, 2022 at 18:57 | comment | added | YCor | By the way let me mention a variant of these kinds of "local-to-global" questions: arxiv.org/abs/1508.02247 arxiv.org/abs/1512.02775 | |
| Feb 13, 2022 at 17:08 | comment | added | HJRW | ... So the examples of hyperbolicity and Property (T) seem to be answers to different questions, with the quantifiers in different orders. By the way, there is a procedure, due to Ozawa, that looks at a finitely presented ball and always determines (on a large enough ball) if the group has (T). arxiv.org/abs/1312.5431 | |
| Feb 13, 2022 at 17:05 | comment | added | HJRW | ... On the other hand, you claim that hyperbolicity is such a property, but this only depends on a finite ball in the Cayley graph which in turn depends on the presentation. But, as @YCor points out, the finitely presented group is determined by that ball up to isomorphism, so in fact it's trivial that every property is determined this way. The interesting thing about the hyperbolicity fact is that this determination is actually computable, but the question seems to avoid computability issues... | |
| Feb 13, 2022 at 17:02 | comment | added | HJRW | I'm confused by this question, which seems to admit several interpretations. If we really only want properties that depend only on a fixed ball in the Cayley graph, then we are talking about properties that are open in the space of marked groups, such as Property (T)... | |
| Feb 13, 2022 at 11:12 | comment | added | ARG | @YCor I agree, you should add the number of generators as an input. | |
| Feb 13, 2022 at 11:09 | comment | added | YCor | My main point is that if you also fix the number of generators, you get finitely many presentations. Otherwise you can get a huge number of generators not appearing in relations (since the sum of all lengths of all relators is bounded). This makes the setting a bit weird (unless this complexity is really desired) | |
| Feb 13, 2022 at 10:58 | comment | added | ARG | good point! It's probably more convenient to make the number of generators an input too (at least for "Betti numbers"-like reason). But then again, I don't know if it's relevant (you can certainly reduce the length of the relations by adding more generators, but then you also increase the number of relations, right?) | |
| Feb 13, 2022 at 10:47 | comment | added | YCor | You fix the number of relations and their length, but not the number of generators? | |
| Feb 13, 2022 at 10:42 | history | asked | ARG | CC BY-SA 4.0 |