Equations in acylindrically hyperbolic groups and verbal closedness

TL;DR

This paper characterizes solutions to specific equations in acylindrically hyperbolic groups and establishes that certain verbally closed subgroups are precisely the retracts, advancing understanding of subgroup structure in these groups.

Contribution

It provides a description of solutions to equations in AH-groups and proves that finitely generated acylindrically hyperbolic subgroups without finite normal subgroups are retracts if and only if they are verbally closed.

Findings

Solutions to $x^ny^m=a^nb^m$ are characterized in AH-groups.

Verbal closedness of subgroups is equivalent to being a retract under certain conditions.

Solved a problem on verbally closed subgroups of hyperbolic groups.

Abstract

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.