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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.

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  • $\begingroup$ A quasi- restatement of the assertion is: if a group $G$ admits an acylindrical action on a quasi-line (=space QI to a line), then $G$ is virtually cyclic. [Well this is slightly stronger since in the original question one assumes that it extends to a non-elementary [[in which sense? is focal allowed?]] action on a larger space] $\endgroup$
    – YCor
    Commented Aug 6, 2021 at 18:11

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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.

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  • $\begingroup$ Thanks for the explanation. I wonder why the action of $H$ has to be elementary so that $H$ doesn't contain infinitely many independent loxodromic elements (condition c of the Theorem 1.1)? $\endgroup$
    – Joseph
    Commented Aug 7, 2021 at 2:09
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    $\begingroup$ A loxodromic isometry has only two fixed points at infinity: the endpoints of a quasi-axis. So, if $H$ contains at least two independent loxodromic isometries, then one of them will not fix $\alpha$ or $\omega$. $\endgroup$
    – AGenevois
    Commented Aug 7, 2021 at 5:58

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