Geodesic
Characteristic properties and uniform non-amenability of $n$-periodic products of groups
Izvestiya. Mathematics , Tome 79 (2015) no. 6, pp. 1097-1110

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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.
Keywords: $n$-periodic product, free periodic group, uniform non-amenability, exponential growth.
Mots-clés : simple group, amenable group 
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     title = {Characteristic properties and uniform non-amenability of $n$-periodic products of groups},
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S. I. Adian; Varuzhan Atabekyan. Characteristic properties and uniform non-amenability of $n$-periodic products of groups. Izvestiya. Mathematics , Tome 79 (2015) no. 6, pp. 1097-1110. http://geodesic.mathdoc.fr/item/IM2_2015_79_6_a0/