a positive solution of the problem of the existence of nontrivial pairs of zero-divisors in group rings of free burnside groups of sufficiently large odd periods $n>10^{10}$ obtained previously by s. v. ivanov and r. mikhailov extended to all odd periods $ngeq 665$‎.

we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.

We prove that any maximal group in the free Burnside semi-group deened by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the J-class containing the group. For n = 2 and for every m 2 we present examples with 2m...

In 1902 Burnside [4] wrote "A still undecided point in the theory of discontinuous groups is whether the order of a group may be not finite while the order of every operation it contains is finite". This leads to the following problem, now called the Burnside problem: "If a group is finitely generated and of finite exponent, is it finite?" This is a very difficult question so a weaker form know...

In a pair of recent articles, the author develops a general version of small cancellation theory applicable in higher dimensions ([5]), and then applies this theory to the Burnside groups of sufficiently large exponent ([6]). More specifically, these articles prove that the free Burnside groups of exponent n ≥ 1260 are infinite groups which have a decidable word problem. The structure of the fi...

The simplest example of an infinite Burnside group arises in the class of automaton groups. However there is no known example of such a group generated by a reversible Mealy automaton. It has been proved that, for a connected automaton of size at most 3, or when the automaton is not bireversible, the generated group cannot be Burnside infinite. In this paper, we extend these results to automata...

We construct an embedding of a free Burnside group B(m, n) of odd n > 2 and rank m > 1 in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another a...