Timeline for Properties of a non-sofic group
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12 events
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| Feb 23, 2011 at 6:49 | comment | added | Denis Osin | @ Andreas: Kate Juschenko showed me your answer after I mentioned exactly the same approach. I tried it few years ago and here is what I found. I wrote a simple program in GAP which randomly choses the way to match pairs of FxK, produces the corresponding group presentation with generators F,K, and simplifies it via Tietze transformations. I then analized about 10^5 random matchings for various cardinalities of F and K. In all cases the resulting groups happened to be obviously sofic (even residually finite), so ba=1 follows. For |F|,|K|-> infinity, the group was Z with probability -> 1. | |
| Oct 29, 2010 at 6:36 | comment | added | Andreas Thom | @Mark: I read the same story in Yann Olliviers survey about random groups. It was Richard Kenyon who performed these experiments and first thought he contradicted Gromov since he also found an omission in Gromov's proof. Later the omission was filled by Ol'shanskii and it was pointed out that taking 500 generators and 2000 relations is still not producing any randomness, at least not the predicted effects. | |
| Oct 28, 2010 at 6:04 | history | edited | Andreas Thom | CC BY-SA 2.5 |
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| Oct 28, 2010 at 0:07 | comment | added | user6976 | If I remember correctly, Gromov told me that somebody (in Orsay?) made a computer experiment with random groups, trying to show that property (T) occurs with probability 1 for density between $1/3$ and $1/2$ (which is proved by Gromov). The results were disappointing because the probability converges very slowly, and one needs huge number of relations (and very large relations) to get probability anywhere close to 1. | |
| Oct 27, 2010 at 21:49 | comment | added | Jon Bannon | For convenience: seven.ihes.fr/~gromov/PDF/6%5B106%5D.pdf (Gromov's random group paper...) | |
| Oct 27, 2010 at 21:07 | comment | added | user6976 | @Jon: Computer won't help because you won't be able to show using a computer that certain relations do not hold. One can try to use Gromov random groups method and assign the matching randomly. In the case of zero divisors, though the "Gromov density" was 1/2, which means we can conclude nothing. I suspect that this is the case here too. | |
| Oct 27, 2010 at 20:20 | comment | added | Jon Bannon | I still think this idea is tantalizing, though. Perhaps if we aimed at it with a computer in order to "stumble" on a counterexample! | |
| Oct 27, 2010 at 19:42 | history | edited | Andreas Thom | CC BY-SA 2.5 |
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| Oct 27, 2010 at 18:38 | history | edited | Andreas Thom | CC BY-SA 2.5 |
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| Oct 27, 2010 at 18:11 | comment | added | user6976 | This is similar to the zero divisors conjecture, and looks as hopeless: too many relatively short relations. | |
| Oct 27, 2010 at 18:00 | history | edited | Andreas Thom | CC BY-SA 2.5 |
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| Oct 27, 2010 at 16:28 | history | answered | Andreas Thom | CC BY-SA 2.5 |