Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $H$. Does $H$ permute the components of $\partial G - \Lambda H?$

Question

Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $G$. In particular, $\partial G$ is connected but $\partial G - \Lambda H$ is disconnected. The number of components of $\partial G - \Lambda H$ can be identified with the number of co-ends of $H$ in $G$ - i.e. the number of ends of the coset graph $H \backslash G$.

It is easy to see that $H$ acts on $\partial G - \Lambda H$ since $H$ acts on $\Lambda H$. My question is then the following.

Does $H$ stabilise each connected component of $\partial G - \Lambda H$, or does $H$ permute these components?

I think if $H$ has infinitely many co-ends in $G$, then $H$ must necessarily permute these components. However, I cannot think of examples with finitely many co-ends where these pieces are not fixed by $H$.

Any illuminating examples would be appreciated. Thanks!

Answer 1

Here is a more natural example that the one I gave in the comments.

Given a graph $\Gamma$ and a collection of groups $\mathcal{G}= \{G_u \mid u \in V(\Gamma) \}$ indexed by the vertices of $\Gamma$, define the graph product $$\Gamma \mathcal{G}:= \langle G_u, \ u \in V(\Gamma) \mid [G_u,G_v]=1, \ \{u,v\} \in E(\Gamma) \rangle.$$ In other words, we take the free product of the $G_u$ and we add relations so that every element in $G_u$ commutes with every element in $G_v$ as soon as $u,v$ are adjacent vertices in $\Gamma$. For instance, one can think about a right-angled Coxeter groups.

If all the groups in $\mathcal{G}$ are finite, then $\Gamma \mathcal{G}$ is hyperbolic iff $\Gamma$ is square-free (i.e. no induced cycle of length four). The Cayley graph $\mathrm{QM}(\Gamma ,\mathcal{G}):= \mathrm{Cayl}(\Gamma \mathcal{G}, \bigcup \mathcal{G} )$ has a very nice geometry: it is a quasi-median graph. Like in median graphs (a.k.a. one-skeleta of CAT(0) cube complexes), there are hyperplanes and hyperplane-stabilisers, which are conjugates of subgroups of the form $\langle \mathrm{star}(u) \rangle$ in our case, are codimension-one subgroups (when they have infinite indices).

In the hyperplane-stabiliser $\langle \mathrm{star}(u) \rangle= \langle \mathrm{link}(u) \rangle \times \langle u \rangle$, the factor $\langle u \rangle$ permutes the sectors delimited by the corresponding hyperplane.

I can add more details if needed.