Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.


* Say that the action is acylindrical if for every $\epsilon$, there exist $R,N$ such that for every two points $x,y\in X$ with $d(x,y)\geq R$, there are at most $N$ elements $g\in \Gamma$ such that both $d(x,g\cdot x)\leq \epsilon$ and $d(y,g\cdot y)\leq \epsilon$.
* Let $h\in \Gamma$ be a loxodromic element with respect to the action. Say that $h$ is WPD if for every $\epsilon$ and every $x\in X$, there exists $m\in \mathbb{N}$ such that the set of elements $g\in \Gamma$ satisfying both $d(x,g\cdot x)\leq \epsilon$ and $d(h^m\cdot x,gh^m\cdot x)\leq \epsilon$ is finite.


Of course if the action is acylindrical, every loxodromic element is WPD.
One of the main result of Osin's paper [Acylindrically hyperbolic groups](https://arxiv.org/abs/1304.1246) is that if $\Gamma$ is not virtually cyclic and acts on a hyperbolic space $X$ with a WPD element, then it acts acylindrically on a hyperbolic space $Y$.




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We thus have two boundaries for $\Gamma$, namely its limit set $\Lambda\_X\Gamma$ in the Gromov boundary of $X$ and its limit set $\Lambda\_Y\Gamma$ in the Gromov boundary of $Y$. My question is as follows.


**Question.** Can we construct the space $Y$ such that $\Lambda\_X\Gamma$ and $\Lambda\_Y\Gamma$ equivariantly homeomorphic ? At least can we construct the space $Y$ such that there is an equivariant embedding $\Lambda\_Y\Gamma\hookrightarrow \Lambda\_X\Gamma$ ? For example, does Osin's construction of $Y$ yield an embedding $\Lambda\_Y\Gamma\hookrightarrow \Lambda\_X\Gamma$ ?




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**Some motivation.** On the one hand, usually, when $\Gamma$ acts on a space $X$ with WPD elements, one has a geometric interpretation of $\Lambda\_X\Gamma$. This is for example the case for the group $Out(F\_n)$ acting on the free factor complex or on the sphere complex. On the other hand, when one has an acylindrical action, one can derive a lot of properties from this action. 


One thing I'm interested in is the following. Given a random walk on $\Gamma$, the set $\Lambda\_Y\Gamma$ endowed with the harmonic measure is a model for the Poisson boundary.


Thus, if $\Gamma$ acts on $X$ with a WPD element and if $\Lambda\_Y\Gamma\hookrightarrow \Lambda\_X\Gamma$ (or even better $\Lambda\_Y\Gamma\simeq \Lambda\_X\Gamma$) then one has a nice geometric interpretation of the Poisson boundary.




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Basically, the proof of Osin's theorem goes as follows. Recall that a subgroup $H$ of $\Gamma$ is hyperbolically embedded in $\Gamma$ (with respect to a subset $S$ of $\Gamma$) if


* the group $\Gamma$ is generated by $S$ and $H$ and the Cayley graph $\mathrm{Cay}(\Gamma, S\cup H)$ is hyperbolic,
* the subgroup $H$, endowed with the induced metric $d\_H$ is a proper metric space. This induced metric $d\_H(h\_1,h\_2)$ is basically given by the smallest possible length of a path from $h\_1$ to $h\_2$ staying outside of $H$.


Now, assume that $\Gamma$ acts on $X$ with a WPD element.
First, if $h$ is WPD, then it is contained in a maximal virtually cyclic subgroup $E(h)$ and $E(h)$ hyperbolically embeds into $\Gamma$. Second, if $H$ is hyperbolically embedded into $\Gamma$ with respect to a set $S$, define a set $S'$ consisting of all elements $g$ such that a geodesic from 1 to $g$ does not stay longer than $D$ inside $H$, for some fixed constant $D$. Then, the Cayley graph $\mathrm{Cay}(\Gamma,S'\cup H)$ is hyperbolic and the action of $\Gamma$ on this Cayley graph is acylindrical.


**Let us look at an easy example now.** Consider the free group $\Gamma=F\_2$ and let $X$ be its Cayley graph with respect to the standard system of generators. Denote by $a$ and $b$ these two generators. Then, $a$ is WPD and one can take $E(a)=\langle a\rangle$. Clearly, $\langle a \rangle$ hyperbolically embeds into $\Gamma$ with respect to the set $S=\{b\}$.


Now, Osin's construction yields $S'=\langle b\rangle$. Thus, the space $Y$ on which $\Gamma$ acylindrically acts is $\mathrm{Cay}(\Gamma,\langle a\rangle \cup \langle b\rangle)$. This Cayley graph $\mathrm{Cay}(\Gamma,\langle a\rangle \cup \langle b\rangle)$ is quasi-isometric to the coned-off graph of $\Gamma$, considered as hyperbolic relative to its free factors. Its Gromov boundary is the set of conical limit points and it embeds into the Gromov boundary of $\Gamma$. Precisely, the boundary of $\mathrm{Cay}(\Gamma,\langle a\rangle \cup \langle b\rangle)$ is the set of infinite words alternating elements of $\langle a \rangle$ and elements of $\langle b \rangle$.


In particular, in this example, we indeed have $\Lambda\_Y\Gamma\hookrightarrow\Lambda\_X\Gamma$. Note however that one could take $Y$ to be the standard Cayley graph of $\Gamma$ and then one would get $\Lambda\_Y\Gamma\simeq\Lambda\_X\Gamma$.



Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
I think the equivariant embedding you ask for is given in Theorem 3.2 of this paper:
<https://arxiv.org/pdf/1601.00101.pdf>. 


Actually, there is such an embedding any time you cone off uniformly quasiconvex subspaces of a hyperbolic space. 


Added:


Theorem 3.2 states that if $X$ and $Y$ are hyperbolic and $f \colon X \to Y$ is coarsely Lipschitz, coarsely surjective, and alignment preserving, then there is a subspace $\partial\_Y X$ of $\partial X$ that is homeomorphic to $\partial Y$. The inverse of the homeomorphism is the embedding $\partial Y \to \partial X$ that you are looking for.


One source of examples of such a map $f \colon X \to Y$ comes from work of Kapovich--Rafi. See Corollary 2.4 of this paper <https://arxiv.org/pdf/1206.3626.pdf>, where the alignment preserving property is exactly their `Moreover' statement. In fact, Corollary 2.4 is the result that Osin uses to show that his space (i.e. $\mathrm{Cay}(\Gamma,S'\cup H)$ in your notation) is hyperbolic.


As for a precise statement about coning off quasiconvex subspaces, see Proposition 2.6 in the Kapovich--Rafi paper above. In that setting, the resulting map $X \to Y$ is alignment preserving and so gives an embedding $\partial Y \to \partial X$ by Theorem 3.2 of my paper with Spencer.


