# Arzelà-Ascoli for hyperbolic spaces with natural boundary

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?

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 $$X$$.