Equations in acylindrically hyperbolic groups and verbal closedness

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 projec-tion of the algebraic set associated with a single splitted equation (such an equation has the form w.x1; : : : ; xn/ D h, where w 2 F .X/, h 2 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 D an 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.