Probabilistic identities in groups and fixed points of automorphisms of periodic groups

In 1959, P. S. Novikov published a note in which he proposed a certain approach to the solution of the Burnside problem for periodic groups. This publication became a major sensation in algebra. However, the problem turned out to be exceedingly difficult, and this approach was finally realized through the monumental joint work of P. S. Novikov and S. I. Adian. Currently, the remarkable theory they created is known as the Novikov–Adian theory. Based on this theory, a number of major problems in group theory have been solved by many of their followers.

In our talk, we will present recent results obtained by us in this field. In particular, we will show how S. I. Adian's $n$-periodic products, constructed on the basis of the Novikov–Adian theory, and an infinite series of independent group identities in two variables help answer the following question: if the probability that a certain identity holds in a group tends to $1$, is it an identity in the usual sense? Furthermore, we will discuss two problems concerning automorphisms of periodic groups posed by V. D. Mazurov in the Kourovka Notebook (2010), which can be solved through a further modification of the Novikov–Adian theory.