Let G be a finitely presented group, and let H be a subgroup of G. We prove that if H is acylindrically hyperbolic and existentially closed in G, then G is acylindrically hyperbolic. As a corollary, any finitely presented group which is existentially equivalent to the mapping class group of a surface of finite type, to Out(Fn) or Aut(Fn) for n ≥ 2 or to the Higman group, is acylindrically hyperbolic.
The following properties are preserved under elementary equivalence, among finitely generated groups: being hyperbolic (possibly with torsion), being hyperbolic and cubulable, and being a subgroup of a hyperbolic group. In other words, if a finitely generated group G has the same first-order theory as a group possessing one of the previous property, then G enjoys this property as well.
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