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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 6, Pages 3–17 (Mi izv8369)  

This article is cited in 4 scientific papers (total in 4 papers)

Characteristic properties and uniform non-amenability of $n$-periodic products of groups

S. I. Adiana, Varuzhan Atabekyanb

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Yerevan State University

Abstract: We prove that $n$-periodic products (introduced by the first author in 1976) are uniquely characterized by certain quite specific properties. Using these properties, we prove that if a non-cyclic subgroup $H$ of the $n$-periodic product of a given family of groups is not conjugate to any subgroup of the product's components, then $H$ contains a subgroup isomorphic to the free Burnside group $B(2,n)$. This means that $H$ contains the free periodic groups $B(m,n)$ of any rank $m>2$, which lie in $B(2,n)$ ([1], Russian p. 26). Moreover, if $H$ is finitely generated, then it is uniformly non-amenable. We also describe all finite subgroups of $n$-periodic products.

Keywords: $n$-periodic product, free periodic group, simple group, amenable group, uniform non-amenability, exponential growth.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-51-05012 Арм_а
15RF-054
This work was carried out with the financial support of the Russian Foundation for Basic Research and the RA MES State Committee of Science in the framework of the joint scientific programme (projects 15-51-05012-Arm\_a and 15RF-054 respectively).


DOI: https://doi.org/10.4213/im8369

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English version:
Izvestiya: Mathematics, 2015, 79:6, 1097–1110

Bibliographic databases:

Document Type: Article
UDC: 512.54+512.543.5
MSC: 20F05, 20F50, 20E06
Received: 25.03.2015
Revised: 16.05.2015

Citation: S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. RAN. Ser. Mat., 79:6 (2015), 3–17; Izv. Math., 79:6 (2015), 1097–1110

Citation in format AMSBIB
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\paper Characteristic properties and uniform non-amenability of $n$-periodic products of groups
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\pages 3--17
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\transl
\jour Izv. Math.
\yr 2015
\vol 79
\issue 6
\pages 1097--1110
\crossref{https://doi.org/10.1070/IM2015v079n06ABEH002774}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71  mathnet  crossref  crossref  isi  elib
    2. S. I. Adian, V. S. Atabekyan, “$C^*$-Simplicity of $n$-Periodic Products”, Math. Notes, 99:5 (2016), 631–635  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. S. Atabekyan, A. L. Gevorgyan, Sh. A. Stepanyan, “The unique trace property of n-periodic products of groups”, J. Contemp. Math. Anal., 52:4 (2017), 161–165  crossref  mathscinet  zmath  isi  elib  scopus
    4. S. I. Adian, V. S. Atabekyan, “Periodic products of groups”, J. Contemp. Math. Anal., Armen. Acad. Sci., 52:3 (2017), 111–117  mathnet  crossref  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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