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Oleg Bogopolski

Publications and source records attributed to Oleg Bogopolski.

18 recordsLinked to original sources
arXiv2025

Finite non-parabolic subgroups of relatively hyperbolic groups

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_λ\}_{λ\in Λ}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We give an upper bound on the orders of finite non-parabolic subgroups of $G$ in terms of some fundamental constants associated with $\mathcal{P}$. This upper bound is computable if $G$ is finitely generated and the word problem in each $H_λ$, $λ\in Λ$, is decidable.

math.GR↗
arXiv2022

Structure of solutions of exponential equations in acylindrically hyperbolic groups

Let $G$ be a group acting acylindrically on a hyperbolic space and let $E$ be an exponential equation over $G$. We show that $E$ is equivalent to a finite disjunction of finite systems of pairwise independent equations which are either loxodromic over virtually cyclic subgroups or elliptic. We also obtain a description of the solution set of $E$. We obtain stronger results in the case where $G$ is hyperbolic relative to a collection of peripheral subgroups $\{H_λ\}_{λ\in Λ}$. In particular, we prove in this case that the solution sets of exponential equations over $G$ are $\mathbb{Z}$-semilinear if and only if the solution sets of exponential equations over every $H_λ$, $λ\in Λ$, are $\mathbb{Z}$-semilinear. We obtain an analogous result for finite disjunctions of finite systems of exponential equations and inequations over relatively hyperbolic groups in terms of definable sets in the weak Presburger arithmetic.

math.GR↗
arXiv2021

Exponential equations in acylindrically hyperbolic groups

Let $G$ be an acylindrically hyperbolic group and $E$ an exponential equation over $G$. We show that if $E$ is solvable in $G$, then there exists a solution whose components, corresponding to loxodromic elements, can be linearly estimated in terms of lengths of the coefficients of $E$. We give a more precise answer in the case where $G$ is a relatively hyperbolic group. Under some assumption of general character, the solvability and the search problems for exponential equations over $G$ can be reduced to the peripheral subgroups of $G$.

math.GR↗
arXiv2021

Notes about decidability of exponential equations

We study relationship among versions of the Knapsack Problem where variables take values in Z and the number of them is fixed. In particular, we construct a finitely presented group where the problem of solvability of exponential equations with one variable is decidable but the corresponding problem for two variables is undecidable.

math.GR↗
arXiv2020

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

We describe homomorphisms $φ:H\rightarrow G$ for which the codomain is acylindrically hyperbolic and the domain is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of $φ$ is small or $φ$ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to $G$ as above. We prove a more precise result for homomorphisms $φ:H\rightarrow {\rm Mod}(Σ)$, where $H$ as above and ${\rm Mod}(Σ)$ is the mapping class group of a connected compact surface $Σ$. In this case there exists an open normal subgroup $V\leqslant H$ such that $φ(V)$ is finite. We also prove the analogous statement for homomorphisms $φ:H\rightarrow {\rm Out}(G)$, where $G$ is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

math.GR↗
arXiv2019

On finite systems of equations in acylindrically hyperbolic groups

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system $S$ of equations with constants from $H$ is equivalent to a single equation. We also show that the algebraic set associated with $S$ is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such equation has the form $w(x_1,\dots,x_n)=h$, where $w\in F(X)$, $h\in H$). From this we deduce the following statement: Let $G$ be an arbitrary overgroup of the above group $H$. Then $H$ is verbally closed in $G$ if and only if it is algebraically closed in $G$. Another corollary: If $H$ is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from $H$ is equivalent to a single equation.

math.GR↗
arXiv2019

Equations in acylindrically hyperbolic groups and verbal closedness

We describe solutions of the equation $x^ny^m=a^nb^m$ in acylindrically hyperbolic groups (AH-groups), where $a,b$ are non-commensurable special loxodromic elements and $n,m$ are integers with sufficiently large common divisor. Using this description and certain test words in AH-groups, we study the verbal closedness of AH-subgroups in groups. A subgroup $H$ of a group $G$ is called verbally closed if for any word $w(x_1,\dots, x_n)$ in variables $x_1,\dots,x_n$ and any element $h\in H$, the equation $w(x_1,\dots, x_n)=h$ has a solution in $G$ if and only if it has a solution in $H$. Main Theorem: Suppose that $G$ is a finitely presented group and $H$ is a finitely generated acylindrically hyperbolic subgroup of $G$ such that $H$ does not have nontrivial finite normal subgroups. Then $H$ is verbally closed in $G$ if and only if $H$ is a retract of $G$. The condition that $G$ is finitely presented and $H$ is finitely generated can be replaced by the condition that $G$ is finitely generated over $H$ and $H$ is equationally Noetherian. As a corollary, we solve Problem 5.2 from the paper arXiv:1201.0497v2 of Miasnikov and Roman'kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts.

math.GR↗
arXiv2019

A periodicity theorem for acylindrically hyperbolic groups

We generalize a well known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization will be used later to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

math.GR↗
arXiv2016

From local to global conjugacy of subgroups of relatively hyperbolic groups

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and $H_2$ is not parabolic. Suppose that $H_2$ is elementwise conjugate into $H_1$. Then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. The minimal length of the conjugator can be estimated. In the case where $G$ is a limit group, it is sufficient to assume only that $H_1$ is a finitely generated and $H_2$ is an arbitrary subgroup of $G$.

math.GR↗
arXiv2016

On subgroup conjugacy separability of hyperbolic QVH-groups

A group $G$ is called subgroup conjugacy separable (abbreviated as SCS) if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. An into-conjugacy version of SCS is abbreviated by SICS. We prove that if $G$ is a hyperbolic group, $H_1$ is a quasiconvex subgroup of $G$, and $H_2$ is a subgroup of $G$ which is elementwise conjugate into $H_1$, then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. As corollary, we deduce that fundamental groups of closed hyperbolic 3-manifolds and torsion-free small cancellation groups with finite $C'(1/6)$ or $C'(1/4)-T(4)$ presentations are hereditarily quasiconvex-SCS and hereditarily quasiconvex-SICS, and that surface groups are SCS and SICS. We also show that the word "quasiconvex" cannot be deleted for at least small cancellation groups.

math.GR↗
arXiv2014

Subgroup conjugacy separability for surface groups

A group $G$ is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. We prove that free groups and the fundamental groups of orientable closed compact surfaces are SCS.

math.GR↗
arXiv2011

Abstract commensurators of solvable Baumslag - Solitar groups

We prove that for any natural n>1, the abstract commensurator group of the Baumslag - Solitar group BS(1,n) is isomorphic to the group of 2 by 2 upper triangular matrices A over rational numbers with A_{11}=1. We also prove that for any finitely generated group G with the unique root property the natural homomorphisms Aut(G)--> Comm(G)--> QI(G) are embeddings.

math.GR↗
arXiv2011

The word problem for some uncountable groups given by countable words

We investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application we give a short proof that these homology groups contain the direct sum of 2^{\aleph_0} copies of \mathbb{Q}. Finally, we show that the fundamental group of Griffith's space contains \mathbb{Q}.

math.GR↗
arXiv2010

On subgroup conjugacy separability in the class of virtually free groups

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.

math.GR↗
arXiv2009

A Magnus theorem for some one-relator groups

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed nonorientable surfaces of genus g>3 possess this property. The analogous result for orientable surfaces of any finite genus was obtained by the first author [Geometric methods in group theory, Contemp. Math, 372 (2005) 59-69].

math.GR↗
arXiv2009

On abstract commensurators of groups

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated, torsion-free group which can be mapped onto Z and which has a finitely generated commensurator.

math.GR↗