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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 1, Pages 207–224 (Mi izv471)  

This article is cited in 10 scientific papers (total in 10 papers)

Markov's theorem and algorithmically non-recognizable combinatorial manifolds

M. A. Shtan'ko


Abstract: We prove the theorem of Markov on the existence of an algorithmically non-recognizable combinatorial $n$-dimensional manifold for every $n\geqslant 4$. We construct for the first time a concrete manifold which is algorithmically non-recognizable. A strengthened form of Markov's theorem is proved using the combinatorial methods of regular neighbourhoods and handle theory. The proofs coincide for all $n\geqslant 4$. We use Borisov's group [8] with insoluble word problem. It has two generators and twelve relations. The use of this group forms the base for proving the strengthened form of Markov's theorem. (The author is indebted to S. I. Adian for this idea.)

DOI: https://doi.org/10.4213/im471

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English version:
Izvestiya: Mathematics, 2004, 68:1, 205–221

Bibliographic databases:

UDC: 510.5+515.164
MSC: 20F10, 57Q40, 57N15
Received: 30.01.2003

Citation: M. A. Shtan'ko, “Markov's theorem and algorithmically non-recognizable combinatorial manifolds”, Izv. RAN. Ser. Mat., 68:1 (2004), 207–224; Izv. Math., 68:1 (2004), 205–221

Citation in format AMSBIB
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\paper Markov's theorem and algorithmically non-recognizable combinatorial manifolds
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\yr 2004
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\pages 207--224
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\pages 205--221
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. A. Shtan'ko, “To the Markov theorem on algorithmic nonrecognizability of manifolds”, J. Math. Sci., 146:1 (2007), 5622–5623  mathnet  crossref  mathscinet  zmath  elib
    2. L. M. Cabrer, D. Mundici, “A Stone-Weierstrass theorem for MV-algebras and unital  -groups”, Journal of Logic and Computation, 2014  crossref  mathscinet  scopus
    3. Coward A., Lackenby M., “An Upper Bound on Reidemeister Moves”, Am. J. Math., 136:4 (2014), 1023–1066  crossref  mathscinet  zmath  isi  scopus
    4. Mundici D., “Invariant Measure Under the Affine Group Over Z”, Comb. Probab. Comput., 23:2 (2014), 248–268  crossref  mathscinet  zmath  isi  scopus
    5. Aschenbrenner M. Friedl S. Wilton H., 3-Manifold Groups, Ems Series of Lectures in Mathematics, Eur. Math. Soc., 2015  crossref  mathscinet  zmath  isi
    6. Mundici D., “A Geometric Approach to MV-Algebras”, On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory, Studies in Fuzziness and Soft Computing, 336, eds. SamingerPlatz S., Mesiar R., Springer-Verlag Berlin, 2016, 57–70  crossref  mathscinet  isi  scopus
    7. Cabrer L.M., Mundici D., “Idempotent endomorphisms of free MV-algebras and unital -groups”, J. Pure Appl. Algebr., 221:4 (2017), 908–934  crossref  mathscinet  zmath  isi  scopus
    8. Lishak B., Nabutovsky A., “Balanced presentations of the trivial group and four-dimensional geometry”, J. Topol. Anal., 9:1 (2017), 15–25  crossref  mathscinet  zmath  isi  scopus
    9. Mundici D., “Fans, Decision Problems and Generators of Free Abelian l-Groups”, Forum Math., 29:6 (2017), 1429–1439  crossref  mathscinet  zmath  isi  scopus
    10. Mundici D., “Recognizing Free Generating Sets of l-Groups”, Algebr. Universalis, 79:2 (2018), UNSP 24  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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