Related papers: Solution to the Burnside Problem

All groups have 2 generators. For every prime power q, the Generalized Burnside Theorem (Theorem GB) produces an infinite number of solvable groups, Some, such as groups of a prime power exponent, have only elements of finite order and are…

All groups are 2-generator. For any prime-power q, Theorem 1 constructs a solvable matrix group over a quotient of a Laurent polynomial ring. This group is closely related to a group of exponent q as shown in Theorems 2 & 3 . Theorem 4 in…

We present a solution to the Burnside Problem for 2 generator groups of prime-power exponent that does not rely on induced maps as in [2]. As before, we construct a surjective map of a rank 2 free group to a solvable group G and finish by…

It is proved that the free $m$-generated Burnside groups $\Bbb{B}(m,n)$ of exponent $n$ are infinite provided that $m>1$, $n\ge2^{48}$.

Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every…

We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a…

We show that the free Burnside groups $B(m,n)$ are infinite for $m\geq 2$ and odd $n\geq 557$, the best currently known lower bound for the exponent. The proof uses iterated small cancellation theory where the induction is based on the…

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

We develop yet another technique to present the free Burnside group $B(m,n)$ of odd exponent $n$ with $m\ge2$ generators as a group satisfying a certain iterated small cancellation condition. Using the approach, we provide a reasonably…

We prove that the word problem of the Brin-Thompson group nV over a finite generating set is coNP-complete for every n \ge 2. It is known that the groups nV are an infinite family of infinite, finitely presented, simple groups. We also…

A group $G$ is said to be periodic if for any $g\in G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $\mu$ is finite. Moreover…

We construct an embedding of a free Burnside group $B(m,n)$ of odd $n > 2^{48}$ 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…

A group $\Gamma$ is said to be periodic if for any $g$ in $\Gamma$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite…

Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent $A(G)$ of finite $p$-groups was obtained in a previous article of the first-named author. In this paper, we compute $A(G)$ for any…

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word…

For a finite group we introduce a particular central extension, the unitary cover, having minimal exponent among those satisfying the projective lifting property. We obtain new bounds for the exponent of the Schur multiplier relating to…

In this paper, we give elementary proofs of the Restricted Burnside Problem and the Hughes Conjecture for finite $p$-groups with Hall's regular power structure property. Moreover, in this setting we determine an explicit bound on the order…

We establish several results on the word problem for just infinite groups. First, for finitely generated just infinite groups we show that the word problem is uniformly decidable for presentations with recursively enumerable sets of…

Let $G$ be a finitely generated group. We prove that the $n$-fold tensor product $G^{\otimes n}$ is finite (resp. polycyclic) if and only $G$ is finite (resp. polycyclic). Further, assuming that $G$ is finitely presented, we show that…