Arzelà-Ascoli for hyperbolic spaces with natural boundary

Question

I am investigating Paulin's method of certain limits of actions on hyperbolic spaces being (in some sense) actions on $\mathbb{R}$-trees.

Let $G$ be a finitely generated group. Part of the proof is saying that given a sequence of non-elementary isometric actions $\rho_i: G \to \textrm{Isom} X$ on a proper, cocompact hyperbolic space $X$ such that there are points $x_i$ which generators of $G$ move by at most uniformly bounded distance, we can find a subsequence of the actions, which after conjugation by an isometry converge to an action $\rho: G \to \textrm{Isom} X$ (let's say they converge pointwise).

My question is as follows. Isometries of $X$ extend naturally to the natural compactification $X \cup \partial X$ ($X$ is a hyperbolic metric space). Therefore the actions (evaluated at a given $g \in G$, say) seem to be calling for some topological version of Arzelà-Ascoli. Does anyone know a version which would be useful in the setting of hyperbolic spaces with the natural boundary?

Answer 1

Here is one:

Theorem. Let $X$ be a proper geodesic $\delta$-hyperbolic space and $f_n: X\to X$ is a sequence of $L$-bilipschitz homeomorphisms such that there exists $x\in X, C\in {\mathbb R}$ satisfying $d(x, f_n(x))\le C$. Then, after extraction, the sequence $$ f_n: \bar{X}\to \bar{X} $$ converges in the uniform topology. Here $\bar{X}=X\cup \partial X$ equipped with the usual topology (and the uniform structure).

A proof is an application of the Morse Lemma and chasing through the definition of the uniform structure on $\bar{X}$. One can prove a similar result by weakening $L$-bilipschitz to $(L,A)$-quasi-isometric, but that requires modifying the notion of convergence on $X$.