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Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free groups and torsion free hyperbolic groups contains mistakes and so, that problem which was announced to be solved in 2006, is still an open problem. At this time, I am interested to know, which important theorems of Group theory, Model theory and Algebraic geometry over groups discovered in the period of 2006-2013 applied result of Kharlampovich-Miyasnikov.

Edit: An answer of Kharlampovich and Miyasnikov for the preprint of Sela is just published in arXiv. They explained briefly that there was no serious mistakes in their work, and many errors discovered by Sela are already have been corrected. See this link: http://arxiv.org/abs/1402.0482

Sh.M1972
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    I guess that a related question that should be answered before is whether most of the mathematical community working on these topics also agree with Sela's point of view. – boumol Jan 24 '14 at 09:37
  • I am not sure, but the preprint of Sela contains many counterexamples. – Sh.M1972 Jan 24 '14 at 09:44
  • @Shahryari: For sure it is said so in the preprint (and for sure Sela is a world-renowned mathematician), but I am not an expert on the topic. Thus, let me rephrase my related question: Has any other expert checked the counterexamples given by Sela (or the proof by Olga and Alexi)? Can some people (besides Sela) working on the field say his opinion? – boumol Jan 24 '14 at 10:04
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    @boumol, MO is absolutely the wrong place to have that discussion - it's far too contentious. On the other hand, the original question is perfectly clear and straightforward - it asks for a list of results that rely on Olga and Alexei's work. – HJRW Jan 24 '14 at 11:40
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    The statement " $Th(F_n)$ is decidable" belongs to the class of those mathematical discoveries which are applicable in very wide areas of mathematics. It is not just a hard theorem in a special branch of algebra. In my opinion ( I am not an expert, but I have a few studies in this field), since 2006 many mathematicians used this result to develop new ideas and clearly if it will be false then many other results should be revised. – Sh.M1972 Jan 24 '14 at 11:58
  • @HJRW: Do you mean that a question perfectly clear and straightforward like "list some consequences of the statement there are non trivial natural number where Fermat-Wiles fails" is worth being discussed at mathoverflow? – boumol Jan 24 '14 at 12:21
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    @boumol I am not sure what point you want to make but see http://mathoverflow.net/questions/140459/has-fermats-last-theorem-per-se-been-used Of course not each and every request of this form will make a good question; for one thing because depending on the result it could be either 'too broad' or 'too localized' or something a long these lines, but certain ones should be reasonable. –  Jan 24 '14 at 13:32
  • @quid: I think you didn't understand my last question, I was wondering of consequences of the fact that "Fermat-Wiles Theorem is false". – boumol Jan 24 '14 at 13:49
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    @boumol: No. The question is fundamentally 'List some consequences of the decidability of the elementary theory of free groups (or hyperbolic groups, if you prefer).' This is an OK question, completely independently of the status of Alexei and Olga's results. Note that the question is NOT: 'List some consequences of the undecidability of the elementary theory of free groups.' – HJRW Jan 24 '14 at 14:10
  • @HJRW: I do not see why you somehow suggest that 'List some consequences of the undecidability of the elementary theory of free groups' is not interesting. – boumol Jan 24 '14 at 14:58
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    @boumol, I didn't say it's not interesting, I just said it's not what the question asks for. It would also be beside the wider point, since no one claims that the theory of free groups is undecidable. – HJRW Jan 24 '14 at 15:03
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    I think the question can be answered in part by using a citation index. – The Masked Avenger Jan 24 '14 at 15:18
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    @TheMaskedAvenger, I doubt this is the case since this result is quite famous and highly quoted when not applied. – Benjamin Steinberg Jan 24 '14 at 16:51
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    I also think that the title and question should be re-edited to remove contentious statements and in particular the phrase "(wrong!!!??)". The question should just ask what results depend on Olga and Alexei's work. – Benjamin Steinberg Jan 24 '14 at 16:54
  • @Benjamin Steinberg: You are right, I was not enough careful on this issue, I did some editions, both in title and in content. – Sh.M1972 Jan 24 '14 at 16:59
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    You still will have troubles getting such a list. For example, the original solution of the isomorphism problem for limit groups by Bumagin, Kharlampovich and Miasnikov uses the work on the Tarskii problems and in particular the work on effective JSJ decompositions, if I recall correctly. Dahmani and Groves later solved the isomorphism problem for toral relatively hyperbolic groups (which includes limit groups) via independent methods that also gives at least some of the effective JSJ results. – Benjamin Steinberg Jan 24 '14 at 17:04
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    Anyway, the new edit is a little better but I prefer to stay away from this question. I am not sure given the context that any form of this question can be appropriate for MO. – Benjamin Steinberg Jan 24 '14 at 17:05
  • @boumol I admit I misread what precisely you said, sorry about that, but also I feel that the main point still stands, along the lines of what HJRW said. –  Jan 24 '14 at 20:15
  • It may be instructive to think of application of decidability in general. For example, Tarski proved that the first-order theory of real-closed fields is decidable, which has lots of geometric applications, see e.g. Joel David Hamkins's answer http://mathoverflow.net/questions/134259/is-the-first-order-theory-with-of-real-numbers-with-addition-and-multiplicat – Igor Belegradek Jan 25 '14 at 20:39

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Sela does not have an objective view of Kharlampovich, Myasnikov's work. We will post a paper dismissing his statements about "fatal mistakes". It just takes time. There are some typos and inessential errors that were fixed in later works. Sela himself has many such mistakes. The objects in the two works (Sela's and our work) are similar but not identical. They are not amenable to a crude direct interpretation; some of our statements would not be true if interpreted via a ``computer translation'' into his language (and vice versa). One example is
Sela's wrong Theorem 7 from his paper 6 on Diophantine Geometry. This theorem describes groups elementarily equivalent to a non-abelian free groups. Sela claims that our Theorem 41 is wrong too. But our theorem is stated using our concept of regular NTQ groups and is correct. This shows that regular NTQ groups are not completely identical to hyperbolic $\omega$-residually free towers. Many of his critical comments resulted from such an exact translation of our concepts into his language. Additional misrepresentations result from not remembering that some statement was made two pages before (such as that we only consider fundamental sequences satisfying first and second restrictions).

The decidability of the elementary theory of a free group is used in the proof of the decidability of the theory of a torsion free hyperbolic group (our recent preprint in the arxiv) and to make quantifier elimination algorithmic. One can use it to approach the theory of a free product of groups with decidable elementary theories (Malcev's problem). One can also use it to deal with Right Angled Artin groups.

The algorithm to find the abelian JSJ decomposition of a limit group was constructed in our paper "Effective JSJ decompositions" that appeared before, it is used in the proof of the decidability of the theory. Actually many Algebraic Geometry over free groups questions are solved algorithmically (see references on pages 508-514), finding irreducible components of finite systems of equations, analogs of elimination and parametrization theorems in classical Algebraic Geometry etc Olga Kharlampovich

Olga Kharlampovich
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  • That is a great news for everybody interested in works of Kharlampovich-Miyasnikov. Both works of Kharlampovich-Miyasnikov and Sela are outstanding achievements of recent years and I hope you can prepare such a paper about "fatal mistakes" as soon as possible. – Sh.M1972 Feb 02 '14 at 03:41
  • @IgorBelegradek: Sorry, I just want to vote and it was accepted by mistake. You are right, because the main question asks about applications of the decidablity theorem of Kharlampovich-Miyasnikov. We are trying to list some important results which used the work of Kharlampovich-Miyasnikov. – Sh.M1972 Feb 02 '14 at 05:44