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 Proceedings of the YSU, Physical and Mathematical Sciences, 2020, Volume 54, Issue 2, Pages 81–86 (Mi uzeru708)

Mathematics

The set of $2$-genereted $C^*$-simple relatively free groups has the cardinality of the continuum

V. S. Atabekyan

Yerevan State University, Faculty of Mathematics and Mechanics

Abstract: In this paper we prove that the set of non-isomorphic $2$-generated $C^*$-simple relatively free groups has the cardinality of the continuum. A non-trivial identity is satisfied in any (not absolutely free) relatively free group. Hence, they cannot contain a non-abelian absolutely free subgroups. The question of the existence of $C^*$-simple groups without free subgroups of rank $2$ was posed by de la Harpe in 2007.

Keywords: relatively free groups, $C^*$-simple group, amenable radical, nonamenable group, reduced $C^*$-algebra of a group

DOI: https://doi.org/10.46991/PYSU:A/2020.54.2.081

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MSC: Primary 20F05; Secondary 20E10, 20E05, 20D45
Revised: 14.08.2020
Accepted:17.08.2020
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Citation: V. S. Atabekyan, “The set of $2$-genereted $C^*$-simple relatively free groups has the cardinality of the continuum”, Proceedings of the YSU, Physical and Mathematical Sciences, 54:2 (2020), 81–86

Citation in format AMSBIB
\Bibitem{Ata20} \by V.~S.~Atabekyan \paper The set of $2$-genereted $C^*$-simple relatively free groups has the cardinality of the continuum \jour Proceedings of the YSU, Physical and Mathematical Sciences \yr 2020 \vol 54 \issue 2 \pages 81--86 \mathnet{http://mi.mathnet.ru/uzeru708} \crossref{https://doi.org/10.46991/PYSU:A/2020.54.2.081}