arXiv subjects

S. V. Ivanov

Publications and source records attributed to S. V. Ivanov.

42 records · Page 3Linked to original sources

arXiv2002

Embedding free Burnside groups in finitely presented groups

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 non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent $n \gg 1$ which are not locally finite.

math.GR↗

arXiv2002

On HNN-extensions in the class of groups of large odd exponent

A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent $n \gg 1$ is given in the following form. Let $Q$ be a group of odd exponent $n > 2^{48}$ and $\mathcal G$ be an HNN-extension of $Q$. If $A \in \mathcal G$ then let $\mathcal F(A)$ denote the maximal subgroup of $Q$ which is normalized by $A$. By $τ_A$ denote the automorphism of $\mathcal F(A)$ which is induced by conjugation by $A$. Suppose that for every $A \in \mathcal G$, which is not conjugate to an element of $Q$, the group $<τ_A, \mathcal F(A)>$ has exponent $n$ and, in addition, equalities $A^{-k} q_0 A^{k} = q_k$, where $q_k \in Q$ and $k =0, 1, ..., [2^{-16}n]$ ($[2^{-16}n]$ is the integer part of $2^{-16}n$), imply that $q_0 \in \mathcal F(A)$. Then the group $Q$ naturally embeds in the quotient $\mathcal G / \mathcal G^n$, that is, there exists an analog of the HNN-extension $\mathcal G$ of $Q$ in the class of groups of exponent $n$.

math.GR↗

arXiv2002

On subgroups of free Burnside groups of large odd exponent

We prove that every noncyclic subgroup of a free $m$-generator Burnside group $B(m,n)$ of odd exponent $n \gg 1$ contains a subgroup $H$ isomorphic to a free Burnside group $B(\infty,n)$ of exponent $n$ and countably infinite rank such that for every normal subgroup $K$ of $H$ the normal closure $ ^{B(m,n)}$ of $K$ in $B(m,n)$ meets $H$ in $K$. This implies that every noncyclic subgroup of $B(m,n)$ is SQ-universal in the class of groups of exponent $n$.

math.GR↗

arXiv2002

On quasivarities of groups and equations over groups

We prove that the quasivariety of groups generated by finite and locally indicable groups does not contain the class of periodic groups. This result is related to (and inspired by) the solvability of equations over groups. The proof uses the Feit-Thompson theorem on the solvability of finite groups of odd order, Kostrikin-Zelmanov results on the restricted Burnside problem and applies techninal details of a recent construction of weakly finitely presented periodic groups.

math.GR↗

arXiv2002

On the asphericity of LOT-presentations of groups

Let $U$ be an arbitrary word in letters $x_1^{\pm 1}, ..., x_m^{\pm 1}$ and $m \ge 2$. We prove that the group presentation $ $ is aspherical. The proof is based upon prior partial results of A. Klyachko and the author on the asphericity of such presentations.

math.GR↗

arXiv2002

Weakly finitely presented infinite periodic groups

A group $G$ given by a presentation $G = < \mathcal A \| \mathcal R >$ is called weakly finitely presented if every finitely generated subgroup of $G$, generated by (images of) some words in $\mathcal A^{\pm 1}$, is naturally isomorphic to the subgroup of a group $G_0 = < \mathcal A_0 \| \mathcal R_0>$, where $\mathcal A_0 \subseteq \mathcal A$, $\mathcal R_0 \subseteq \mathcal R$ are finite, generated by (images of) the same words. In the article, weakly finitely presented periodic groups which are not locally finite are constructed.

math.GR↗