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 Tr. Mat. Inst. Steklova, 2015, Volume 289, Pages 41–82 (Mi tm3625)

New estimates of odd exponents of infinite Burnside groups

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: This article consists of two parts. The first part presents a detailed history of the long-term joint work (1960–1968) of the author and P.S. Novikov on the proof of the infiniteness of the free Burnside groups $\mathbf {B}(m,n)$ for odd periods $n\ge 4381$ and $m>1$ generators (Sections 1 and 2). In Sections 3–10 we survey several significant results obtained by the author and his successors using the Novikov–Adian theory and its various modifications. In the second part (Sections 11–15) we outline a new modification of the Novikov–Adian theory. The new modification allows us to decrease to $n \ge 101$ the lower bound on the odd periods $n$ for which one can prove the infiniteness of the free periodic groups $\mathbf {B}(m,n)$. We plan to publish a full proof of this new result in the journal Russian Mathematical Surveys.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968515020041

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 289, 33–71

Bibliographic databases:

Document Type: Article
UDC: 512.54

Citation: S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Selected issues of mathematics and mechanics, Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov, Tr. Mat. Inst. Steklova, 289, MAIK Nauka/Interperiodica, Moscow, 2015, 41–82; Proc. Steklov Inst. Math., 289 (2015), 33–71

Citation in format AMSBIB
\Bibitem{Adi15} \by S.~I.~Adian \paper New estimates of odd exponents of infinite Burnside groups \inbook Selected issues of mathematics and mechanics \bookinfo Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov \serial Tr. Mat. Inst. Steklova \yr 2015 \vol 289 \pages 41--82 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3625} \crossref{https://doi.org/10.1134/S0371968515020041} \elib{http://elibrary.ru/item.asp?id=23738462} \transl \jour Proc. Steklov Inst. Math. \yr 2015 \vol 289 \pages 33--71 \crossref{https://doi.org/10.1134/S0081543815040045} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000358577300004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84929952121} 

• http://mi.mathnet.ru/eng/tm3625
• https://doi.org/10.1134/S0371968515020041
• http://mi.mathnet.ru/eng/tm/v289/p41

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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. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110
2. V. S. Atabekyan, “The automorphisms of endomorphism semigroups of free Burnside groups”, Int. J. Algebr. Comput., 25:4 (2015), 669–674
3. S. I. Adian, V. S. Atabekyan, “$C^*$-Simplicity of $n$-Periodic Products”, Math. Notes, 99:5 (2016), 631–635
4. L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099
5. S. I. Adian, V. S. Atabekyan, “On free groups in the infinitely based varieties of S. I. Adian”, Izv. Math., 81:5 (2017), 889–900
6. V. S. Atabekyan, H. T. Aslanyan, A. E. Grigoryan, “Normal automorphisms of free Burnside groups of period 3”, Armen. J. Math., 9:2 (2017), 60–67
7. S. I. Adian, V. S. Atabekyan, “Periodic products of groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117
8. V. S. Atabekyan, A. L. Gevorgyan, Sh. A. Stepanyan, “The unique trace property of $n$-periodic product of groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:4 (2017), 161–165
9. V. S. Atabekyan, H. T. Aslanyan, “The automorphisms of endomorphism semigroups of relatively free groups”, Int. J. Algebr. Comput., 28:2 (2018), 207–215
10. S. I. Adian, “On the studies of Gennadii Semënovich Makanin on algorithmic questions of the theory of groups and semigroups”, Russian Math. Surveys, 73:3 (2018), 553–568
11. S. I. Adian, V. S. Atabekyan, “Central extensions of free periodic groups”, Sb. Math., 209:12 (2018), 1677–1689
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