Given a group $G$, suppose $G$ admits a non-elementary acylindrical action
on a Gromov hyperbolic space $S$.


I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically hyperbolic group is either finite or virtually cyclic but couldn't find a reference. I wonder if someone knows where it is and could tell me.



The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic
I do not know a reference where this statement exactly is proved, but Theorem 1.1 from Osin's article *Acylindrically hyperbolic groups* does most of the work.


It implies that, if the stabiliser $H$ of a pair of points at infinity $\alpha,\omega \in \partial S$ is not virtually infinite cyclic, then it has bounded orbits. But $H$ already quasi-preserves a (quasi-)geodesic $\gamma$ between $\alpha$ and $\omega$. Some basic hyperbolic geometry then implies that $H$ actually quasi-fixes $\gamma$, and it follows from the acylindricity condition that $H$ must be finite.


