Property P naive for acylindrically hyperbolic groups

Abstract

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the P naive property: for any finite collection of elements h 1 , ⋯ , h k , there exists another element γ≠ 1 such that for all i, ⟨ h i , γ⟩ = ⟨ h i ⟩ ∗ ⟨ γ⟩. We also show that if a collection of subgroups H 1 , ⋯ , H k is a hyperbolically embedded collection, then there is γ≠ 1 such that for all i, ⟨ H i , γ⟩ = H i ∗ ⟨ γ⟩.