Property $P_{naive}$ for acylindrically hyperbolic groups

TL;DR

This paper establishes the $P_{naive}$ property for acylindrically hyperbolic groups without non-trivial finite normal subgroups, showing they can freely generate new subgroups with specific elements.

Contribution

It proves the $P_{naive}$ property for a broad class of acylindrically hyperbolic groups and extends this to hyperbolically embedded subgroups, advancing understanding of their subgroup structures.

Findings

Acylindrically hyperbolic groups satisfy the $P_{naive}$ property.

Existence of elements that generate free products with given elements.

Extension to hyperbolically embedded subgroups.

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 $\gamma\neq 1$ such that for all $i$, $\langle h_i, \gamma \rangle = \langle h_i \rangle* \langle \gamma \rangle$. We also obtain that if a collection of subgroups $H_1, \dots, H_k$ is a hyperbolically embedded collection, then there is $\gamma \neq 1$ such that for all $i$, $\langle H_i, \gamma \rangle = H_i * \langle \gamma \rangle$.