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

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.