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 Mat. Zametki, 2020, Volume 107, Issue 5, Pages 752–759 (Mi mz12253)

Residual Nilpotence of Groups with One Defining Relation

D. I. Moldavanskii

Ivanovo State University

Abstract: All groups in the family of Baumslag–Solitar groups (i.e., groups of the form $G(m,n)=\langle a, b; a^{-1}b^ma=b^n \rangle$, where $m$ and $n$ are nonzero integers) for which the residual nilpotence condition holds if and only if the residual $p$-finiteness condition holds for some prime number $p$ are described. It has turned out, in particular, that the group $G(p^r,-p^r)$, where $p$ is an odd prime and $r\ge1$, is residually nilpotent, but it is residually $q$-finite for no prime $q$. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual $p$-finite for some prime $p$, which was announced in the same paper of McCarron, is also given.

Keywords: residual nilpotence, residual $p$-finiteness, one-relator groups.

DOI: https://doi.org/10.4213/mzm12253

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English version:
Mathematical Notes, 2020, 107:5, 820–825

Bibliographic databases:

UDC: 512.543
Revised: 25.04.2019

Citation: D. I. Moldavanskii, “Residual Nilpotence of Groups with One Defining Relation”, Mat. Zametki, 107:5 (2020), 752–759; Math. Notes, 107:5 (2020), 820–825

Citation in format AMSBIB
\Bibitem{Mol20} \by D.~I.~Moldavanskii \paper Residual Nilpotence of Groups with One Defining Relation \jour Mat. Zametki \yr 2020 \vol 107 \issue 5 \pages 752--759 \mathnet{http://mi.mathnet.ru/mz12253} \crossref{https://doi.org/10.4213/mzm12253} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4092570} \elib{https://elibrary.ru/item.asp?id=43284246} \transl \jour Math. Notes \yr 2020 \vol 107 \issue 5 \pages 820--825 \crossref{https://doi.org/10.1134/S0001434620050090} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000542631800009} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85086781217} 

• http://mi.mathnet.ru/eng/mz12253
• https://doi.org/10.4213/mzm12253
• http://mi.mathnet.ru/eng/mz/v107/i5/p752

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This publication is cited in the following articles:
1. V. G. Bardakov, M. V. Neshchadim, “Lower central series of Baumslag–Solitar groups”, Algebra and Logic, 59:4 (2020), 281–294
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