Equations in acylindrically hyperbolic groups and verbal closedness

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.