Acylindrical hyperbolicity and existential closedness

Let G G be a finitely presented group, and let H"> H encoding="application/x-tex">H subgroup of . We prove that if is acylindrically hyperbolic existentially closed in , then hyperbolic. As corollary, any group which equivalent to the mapping class surface finite type, alttext="normal upper O normal u t left-parenthesis F Subscript n Baseline right-parenthesis"> O mathvariant="normal">u mathvariant="normal">t ( F n stretchy="false">) encoding="application/x-tex">\mathrm {Out}(F_n) or A mathvariant="normal">A {Aut}(F_n) for alttext="n greater-than-or-equal-to 2"> ? 2 encoding="application/x-tex">n\geq 2 Higman