A paradoxical route from hyperbolic geometry to proper proximality
For every integer $n\ge 2$, we introduce a new combinatorial condition PP$(n)$ inspired by paradoxical decompositions of groups. As $n$ increases, the property PP$(n)$ weakens, and the resulting hierarchy interpolates between group-theoretic manifestations of negative curvature and proper proximality. More precisely, we prove that a countable group is properly proximal if and only if it satisfies PP$(n)$ for some $n$. On the other hand, every acylindrically hyperbolic group satisfies PP$(2)$; furthermore, finitely generated PP$(2)$-groups admit a geometric characterization: they are precisely the groups containing strongly quasi-convex, non-cyclic, free subgroups. In particular, we obtain that every countable acylindrically hyperbolic group is properly proximal. Finally, for every $n\in\mathbb N$, we provide examples of finitely generated properly proximal groups that do not satisfy PP$(n)$, thus proving that our hierarchy is indeed infinite.