Equations in acylindrically hyperbolic groups and verbal closedness

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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,\ldots,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^ny^m=a^nb^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.