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!



The stabilizers of the canonical boundary action of hyperbolic groups
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](https://homepages.warwick.ac.uk/~masgak/papers/rhg.pdf). 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](https://arxiv.org/pdf/1304.1246.pdf) 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 $P\_k$.


