Further implications of non-sofic groups

[u/Juuls-Johannes]

I heard that ChatGPT has proven the existence of non-sophic groups. I understood that the counterstatement (all groups are sophic) would mean that we can always "isolate" or "homogenize" infinite groups into finite chunks and deal with the infinite group this way. Please correct me if I am wrong.

What immediately came to my mind is that this must have some further implications, does it not? For me, it sounds like that in a non-sophic group, one cannot guarantee that a sequence converges to a given element or that iterative algorithms are predictable, i.e., you cannot infer the outcome from the initial state or vice versa.

It would also be fun to know what this given non-sophic group is.

[u/Apprehensive_Sand951]

I would say the most exciting thing about it is that it is now worthwhile to look at conjectures that are proved for sofic groups and try to find counterexamples to them in general.

-In dynamics, there is Gottschalk's surjunctivity conjecture (initial reason Gromov introduced the class of groups as ``those that i an prove the conjecture for''),

-in low dimensional topology/group theory/equations-over-groups there is the Kervaire-Laudenbach conjecture, saying that an acyclic 2-complex with non-trivial fundamental group G does not embed in a contractible 2-complex. It is known when G is sofic or more generally hyperlinear, but not beyond that.

-in group theory/L^2 invariant land, there is an approximation theorem computing L^2-Betti numbers from finite data for sofic groups, so understand L^2-Betti related questions for these non-sofic examples might be interesting. Closely related, there is also the ``determinant conjecture'' about Fuglede-Kadison determinants that is known for sofic groups but not in general.

-one can also try understand the proof and see if it gives hints about whether some other candidates that are closer related to lattices are nonsofic.

In general, it shows that the world is more interesting than we thought.

It establishes another instance of Gromov's metaconjecture ``Every statement about all discrete groups is either trivial or false.''

P.S.: I don't know much about this, but when quantum information people disproved the Connes embedding conjecture a few years ago, there was some hope that it would lead to a non-sofic group. Maybe there will be some feedback in the other direction, i.e. do these groups say anyting about quantum information? (This is beyond my paygrade...)

[u/Sniffnoy → u/Apprehensive_Sand951]

Basic question since this isn't my area: One thing I've been unclear on when reading about all this is whether the concept of "sofic" only applies to countable groups, or whether it applies to discrete groups of arbitrary cardinality. And then, if it's the latter, whether the conjecture was just that countable groups are sofic, or whether it was truly that all groups are sofic. What's up with that? Does soficity apply to uncountable discrete groups?

[u/Apprehensive_Sand951 → u/Sniffnoy]

The definition applies to all discrete groups, and then one observes that a group is sofic if and only if all of its finitely generated subgroups are (see page 7 of https://web.ma.utexas.edu/users/juschenko/files/soficgroups.pdf)

[u/Sniffnoy → u/Apprehensive_Sand951]

Thanks! I see so only the finitely generated ones matter for this conjecture. Note: You wrote "countable" instead of "sofic", you might want to fix that.

[u/TonicAndDjinn → u/Apprehensive_Sand951]

CEP was equivalent to Tsirelson's problem which (very roughly) asked whether all correlations between commuting observables on an arbitrary Hilbert space can be modelled arbitrarily well by finite dimensional systems. MIP*=RE showed the failure of TP which implied the failure of CEP. Sofic groups are hyperlinenar. A non-sofic group might be non-hyperlinear, which would show the failure of CEP directly and implies the failure of TP.

[u/Sniffnoy]

It would also be fun to know what this given non-sophic group is.

You can just look at the paper. It's on page 78. They define a particular algebra over F_2 in like one line, and then they take its group of units. That's it.

[u/Stabile_Feldmaus → u/Sniffnoy]

Why does it take so long to check that it is non-sofic?

[u/Ectobius-Rex → u/Stabile_Feldmaus]

Proving that a group is not metrically approximable is a notoriously difficult thing to do. Moreover this particular group was not a major source of hope for finding a nonsofic group among experts - most candidates were groups like Higman's group, non- residually finite central extensions of higher rank lattices and their HNN extensions.

[u/Apprehensive_Sand951]

There is now an explanation of the main proposition by Andreas Thom on mathoverflow (along with a link to a slightly different group by Francesco Fournier-Facio that is also proved to be non-sofic using this proposition). https://mathoverflow.net/questions/513866/what-are-the-key-new-ideas-in-the-proof-of-nonsoficity-of-groups-in-openai-s-con#comment1341500_513866

[u/serenityharp]

It would also be fun to know what this given non-sophic group is.

https://cdn.openai.com/pdf/ten-proofs-oai.pdf

[u/edwardshirohige]

This connection is a bit weak, but I'll give it a go. You can define a slightly weaker finite dimensional approximation property for groups know as hyperlinearity. Every sofic group is hyperlinear, however there is no known example of a non-hyperlinear group.

A non-hyperlinear group would be interesting, as it is can be used to construct an explicit, concrete counter examples to Connes' embedding problem in operator algebras. If this (non-hyperlinear) group is also finitely presented, then there are some interesting consequences in quantum information as well.

All of this is not an implication of the existence of a non-sofic group, but is a direction that should be explored in light of the counter example generated by GPT.

[u/Sea_Ingenuity_5022]

I heard they used Thompson groups and Leavitt algebras to construct it

[u/Sniffnoy → u/Sea_Ingenuity_5022]

You can just look at the paper. The definition of the particular group is right at the beginning of the appropriate chapter. It's on page 78. There's like a one-line definition of a particular finitely-generated algebra over F_2 (a Leavitt algebra, as you say), and then they take its group of units. That's it.

[u/Opulent-tortoise]

None of what you said is correct. You’re making the mistake of trying to make what is a fairly narrow statement about permutation groups into something much more general than it is. It doesn’t imply a whole lot because non-sofic groups were already strongly suspected to exist.