Equations in acylindrically hyperbolic groups and verbal closedness

Abstract

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 an 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 $G$.

These statements have interesting implications; here we give only two of them: 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. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29–40] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts.

Moreover, we describe solutions of the equation $x^n{y}^m=a^n{b}^m$ in acylindrically hyperbolic groups (AH-groups), where $a$, $b$ are non-commensurable jointly special loxodromic elements and $n,m$ are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups.