Let be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system of equations with constants from is equivalent to a single equation. We also show that the algebraic set associated with is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form , where , ).
From this we deduce the following statement: Let be an arbitrary overgroup of the above group . Then is verbally closed in if and only if it is algebraically closed in .
These statements have interesting implications; here we give only two of them: If is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from 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 in acylindrically hyperbolic groups (AH-groups), where , are non-commensurable jointly special loxodromic elements and 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.
Cite this article
Oleg Bogopolski, Equations in acylindrically hyperbolic groups and verbal closedness. Groups Geom. Dyn. 16 (2022), no. 2, pp. 613–682DOI 10.4171/GGD/661