Related papers: On the Burnside problem on periodic groups

We construct novel examples of finitely generated groups that exhibit seemingly-contradicting probabilistic behaviors with respect to Burnside laws. We construct a finitely generated group that satisfies a Burnside law, namely a law of the…

In this expository note we provide a proof of Artin's theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the…

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…

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…

Let $P_k$ be the subgroup generated by $k$th powers of primitive elements in $F_r$, the free group of rank $r$. We show that $F_2/P_k$ is finite if and only if $k$ is $1$, $2$, or $3$. We also fully characterize $F_2/P_k$ for $k = 2,3,4$.…

We prove that the outer automorphism group $\mathrm{Out}(N)$ of an infinitely generated free nilpotent group $N$ of class two is complete.

Let $m\geq 3$ be a positive integer. We prove that there are uncountably many non-commensurable metabelian uniform pro-$p$ groups of dimension $m$. Consequently, there are uncountably many non-commensurable finitely presented pro-$p$ groups…

We prove that if the order of a splitting automorphism of free Burnside group~$B(m,n)$ of odd period~$n\ge1003$ is a prime power, then the automorphism is inner. Thus, we give an affirmative answer to the question on the coincidence of…

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non--cyclic free group, and $g$ is an element of $H$, then the…

We prove that, for every integer $n \ge 2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, i.e., every…

It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can…

We consider the set $$\mathcal{A} = \left\{10\cdot a + 11\cdot b \ | \gcd(a,b)=1, a\geq 1, b\geq 2a+1 \right\}.$$ We will prove that $\mathcal{A}$ is unbounded and that there exists a natural number $M\notin \mathcal{A}$ for which…

We show that there exist non-unitarizable groups without non-abelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with non-vanishing…

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 the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…

It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of…

We describe a new class of groups of Burnside type, giving a procedure transforming an arbitrary non-free minimal action of the dihedral group on a Cantor set into an orbit-equivalent action of an infinite finitely generated periodic group.…

We prove that the finiteness of a finitely generated category of irreducible algebraic varieties over a field of characteristic zero is decidable. We also obtain a Burnside finiteness criterion for such a category, with applications to…

Let $m$ and $k$ be integers such that $|m|, \, |k| >1$ and $\gcd (m,k)=1$. We show that all Baumslag-Solitar groups $BS(m,mk)$ are non-residually finite groups hyperbolic relative to residually finite subgroups. By a result of Osin (2007),…