Periodic products of groups

Abstract

In this paper we provide an overview of the results relating to the n-periodic products of groups that have been obtained in recent years by the authors of the present paper, as well as some results obtained by other authors in this direction. The periodic products were introduced by S.I. Adian in 1976 to solve the Maltsev’s well-known problem. It was shown that the periodic products are exact, associative and hereditary for subgroups. They also possess some other important properties such as the Hopf property, the C*-simplicity, the uniform non-amenability, the SQ-universality, etc. It was proved that the n-periodic products of groups can uniquely be characterized by means of certain quite specific and simply formulated properties. These properties allow to extend to n-periodic products of various families of groups a number of results previously obtained for free periodic groups B(m, n). In particular,we describe the finite subgroups of n-periodic products, Also, we analyze and extend the simplicity criterion of n-periodic products obtained previously by S.I. Adian.