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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

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    $\begingroup$ @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. $\endgroup$
    – HJRW
    Commented Jan 24, 2014 at 11:40
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    $\begingroup$ @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.' $\endgroup$
    – HJRW
    Commented Jan 24, 2014 at 14:10
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    $\begingroup$ @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. $\endgroup$
    – HJRW
    Commented Jan 24, 2014 at 15:03
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    $\begingroup$ @TheMaskedAvenger, I doubt this is the case since this result is quite famous and highly quoted when not applied. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 24, 2014 at 16:51
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    $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 24, 2014 at 16:54

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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

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  • $\begingroup$ 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. $\endgroup$
    – Sh.M1972
    Commented Feb 2, 2014 at 3:41
  • $\begingroup$ @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. $\endgroup$
    – Sh.M1972
    Commented Feb 2, 2014 at 5:44

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