I am reading the paper Un groupe hyperbolique est déterminé par son bord, and I am stuck on theorem 3.2, which says the following:
Theorem 3.2: Given two quasi-homogeneous hyperbolic spaces, $X$ and $X'$, the extension to the boundary of a quasi-isometry $f$ from $X$ to $X'$ is an $I$-quasi-conformal homeomorphism.
Proof: If $x'$ is a base point in $X'$, then for any isometry $a_i$ of $X$, it is always possible to find an isometry $b_i$ of $X'$ such that the application $b_i∘f∘a_i$, which is quasi-isometric with uniform constants independent of $i$, moves the point $x'$ by less than a constant independent of $i$. Such quasi-isometries form a compact subset of mappings from $X$ to $X'$ with the topology of uniform convergence on compacts (by the Ascoli's theorem). Therefore, their extensions to the boundary form a compact subset (in the compact-open topology) of homeomorphisms from $\partial X$ to $\partial X'$.
I specifically don't understand why these quasi-isometries form a compact subset of mappings. How can I use Ascoli's theorem to show this?
In the quoted passage the author (F.Paulin) appears to claim the following:
Let $X, X'$ be quasihomogeneous Gromov-hyperbolic spaces, $x\in X, x'\in X'$ be base-points, and let $R, L, A$ be fixed constants. Then the collection $F_{R,L,A}$ of $(L,A)$-quasi-isometries $f: X\to X'$, $d(f(x), x')\le R$, is compact with respect to the topology of uniform convergence on compacts.
This claim is false even if one assumes that $X, X'$ themselves are compact. For instance, take $X=X'=[0,1]$ (with the standard metric). Consider the sequence of maps $f_n: X\to X'$, $f_n(t)=0, t\ne \frac{1}{n}, f_n(1/n)=1$. These maps are $(1,1)$-quasi-isometric. But it is easy to see that the sequence $(f_n)$ contains no subsequence converging in the uniform norm. This is essentially the standard example one learns in a real analysis class during the first encounter with the notion of uniform convergence and its difference from the notion of pointwise convergence.
It is also possible that by "such quasi-isometries" the author has meant ones of the form $f_n=b_n\circ f\circ a_n$, where $a_n, b_n$ are self-isometries of $X, X'$ respectively and $f$ is a fixed quasi-isometry. One finds a counter-example to this claim as well. Namely, take $X=X'={\mathbb R}$ (again with the standard metric). Define $f: X\to X'$ by $f(t)=t$, for all $t$ which is not of the form $n+ \frac{1}{n}, n\in {\mathbb Z}$ and $f(n+ \frac{1}{n})=n+1$ otherwise. Take $a_n(t)=t-n=b_n(t)$. I will leave the details to you to work out.
The truth is that the author has done himself and the reader disservice by using the wrong notion of convergence, inappropriate in the context of coarse geometry. Instead he should have used a coarse form of convergence defined, for instance, by restricting quasi-isometries to a suitable separated net in $X$. With this notion in mind, one can use the Arzela-Ascoli compactness arguments. You can find a detailed discussion for instance in the book by Drutu and Kapovich on Geometric Group Theory.