THE SET OF 2-GENERETED $C^*$-SIMPLE RELATIVELY FREE GROUPS HAS THE CARDINALITY OF THE CONTINUUM - Atabekyan, V.S. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2020)
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; non-amenable group; reduced $C^*$-algebra of a group