Characteristic properties and uniform non-amenability of n-periodic products of groups

摘要

We prove that n-periodic products (introduced by the first author in 1976) are uniquely characterized by certain quite specific properties. Using these properties, we prove that if a non-cyclic subgroup H of the n-periodic product of a given family of groups is not conjugate to any subgroup of the product's components, then H contains a subgroup isomorphic to the free Burnside group B(2, n). This means that H contains the free periodic groups B(m, n) of any rank m > 2, which lie in B(2, n) ([1], Russian p. 26). Moreover, if H is finitely generated, then it is uniformly non-amenable. We also describe all finite subgroups of n-periodic products.