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 $h_1,\dots ,h_k$, there exists another element $\gamma \ne 1$ such that for all $i$, $⟨h_i,\gamma ⟩=⟨h_i⟩\ast ⟨\gamma ⟩$. We also obtain that if a collection of subgroups $H_1,\dots ,H_k$ is a hyperbolically embedded collection, then there is $\gamma \ne 1$ such that for all $i$, $⟨H_i,\gamma ⟩=H_i\ast ⟨\gamma ⟩$.

New sections added with additional results. To appear in Math. Z

References in corpus (3)

• C*-simplicity and the unique trace property for discrete groups
• Non-noetherian groups and primitivity of their group algebras
• Primitivity of group rings of non-elementary torsion-free hyperbolic groups

Cited by in corpus (3)

• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra
• Monodromy kernels for strata of translation surfaces