The stabilizers of the canonical boundary action of hyperbolic groups My question is that 
Is every stabilizer of the canonical boundary action of a hyperbolic group on its Gromov boundary a finitely generated group?
I guess every stabilizer is a (finitely generated) virtually cyclic group, but I do not have a proof nor a reference.
More generally, let G be a countable group that is relatively hyperbolic to subgroups $P_1,\ldots, P_n$. Under which conditions we can conclude that 
every stabilizer of the canonical boundary action on its boundary is a (finitely generated) virtually cyclic group or finitely generated virtually abelian group?
Please see section 2 in https://arxiv.org/pdf/1502.04834.pdf for relative hyperbolic groups and their boundaries.
Thank you very much!
 A: This is following my comment, which was getting too long.
Using the notations of the paper you are citing, there are two kinds of points in the boundary : elements of the Gromov boundary $\partial \Gamma$ of the fine graph $\Gamma$ on which $G$ acts and elements in $V_\infty$, which are vertices of $\Gamma$ of infinite valence. The former are called conical limit points and the later are called parabolic limit points.
This boundary equivariantly agrees with the Gromov boundary $\partial X$ of any proper Gromov hyperbolic space on which $G$ acts via a geometrically finite and minimal action (if you want to remove the word minimal, you need to take the limit set $\Lambda G$ of $G$ instead of $\partial X$). This is Proposition 9.1 combined with Theorem 9.4 in Bowditch's paper relatively hyperbolic groups. In particular, you can choose any fine graph $\Gamma$ on which $G$ acts which satisfies Bowdtich's Definition 2 (which is the definition in the paper you're referring to).
To simplify the following, choose $\Gamma$ to be the coned-off graph with respect to the parabolic subgroups $P_1,...,P_n$, or if you prefer Osin's formulation, the Cayley graph $\mathrm{Cay}(G,S\cup P_1\cup ... \cup P_n)$, where $S$ is any finite generating set, which is quasi-isometric to the coned-off graph. Then, the action of $G$ on this graph $\Gamma$ is acylindrical (this is Proposition 5.2 in Osin's paper that AGenevois indicated in their comment).
Now take any point $\xi$ in the Gromov boundary of $\Gamma$ and let $H$ be its stabilizer. Then, the action of $H$ on $\Gamma$ also is acylindrical and $H$ cannot contain infinitely many independent loxodromic elements, so it is virtually cyclic, by Theorem 1.1 of Osin's paper. This settles conical limit points. On the other hand, let $\xi$ be a parabolic limit point. By point (3) of Bowditch's Definition 2, the stabilizer of $\xi$ is exactly one of the peripheral subgroups, that is, with our notations, is one of the conjugates of the $P_k$.
