# 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!

• Yes (to your first question), it's virtually cyclic. Indeed, it has a loxodromic element (every infinite subgroup of a hyperbolic group has a loxodromic, = infinite order, element). So, if infinite, its action at infinity is axial or focal. In both case, it's quasi-convex. Focal is not possible in this context (it would be if you were considering locally compact compactly generated groups). I'll leave to others to write this in a more standard language (although my wording is more faithful to Gromov's original approach). – YCor Apr 11 at 17:37
• Hyperbolic dynamics is unrelated to hyperbolic groups (at least not in an obvious way), I removed the tag. – YCor Apr 11 at 17:38
• Thank you Yves. Is it possible to provide me a reference? – m07kl Apr 11 at 17:48
• @m07kl If I understand your questions correctly, an answer follows from the classification of acylindrical actions on hyperbolic spaces; see Theorem 1.1 in Osin's paper arxiv:1304.1246. – AGenevois Apr 12 at 9:24
• @AGenevois: Thank you and I will take a look into it – m07kl Apr 12 at 15:00

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 $$P_k$$.
• @m07kl You're welcome and yes you're right ! There is one small thing missing in my argument which I had implicitely in mind. Theorem 1.1 of Osin's paper tells you that the stabilizer of a conical limit point either is virtually cyclic or has bounded orbits on the coned-off graph. To conclude that the group is finite in the later case you need to say that if an element acts with bounded orbits on the coned-off graph, then for the action on the space $X$, it is either eliptic (so has finite order) or is parabolic (which cannot happen here since it already fixes a conical limit point). – M. Dus Apr 14 at 7:42
• @m07kl By the way, if you're interested in groups that are hyperbolic relative to infinitely generated subgroups, you might take a look at Gerasimov and Potyagailo's paper arxiv.org/abs/1008.3470. I'm not used to this framework, but I'm pretty sure that the result in my answer still holds : stabilizers are either virtually cyclic or are a conjugate of one of the $P_k$. – M. Dus May 14 at 12:36