On free groups in the infinitely based varieties of S. I. Adian

We study the free groups in varieties defined by an arbitrary set of identities in a well-known infinite independent system of identities in two variables constructed by S. I. Adian to solve the finite basis problem in group theory. We prove that the centralizer of any non-identity element in a relatively free group in any of the varieties under consideration is cyclic, and for every $m>1$ the set of all non-isomorphic free groups of rank $m$ in these varieties is of the cardinality of the continuum. All these groups have trivial centre, all their abelian subgroups are cyclic, and all their non-trivial normal subgroups are infinite. For any free group $\Gamma$ in any of these varieties, we also obtain a description of the automorphisms of the semigroup $\operatorname{End}(\Gamma)$, answering a question posed by Plotkin in 2000. In particular, we prove that the automorphism group of any such $\operatorname{End}(\Gamma)$ is canonically embedded in the group $\operatorname{Aut}(\operatorname{Aut}(\Gamma))$.