Obtaining a quasi-isometry of the 'boundary' It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends is 'too rough'.
However is was wondering if there is some concept of 'boundary' which is 'fine enough' such that the converse holds or at least holds under some additional assumptions.
If one restricts to finitely generated groups and the attempt to classify them up to quasi-isometry we know that the one-ended groups are the big problem. So probably restricting to one ended groups has there any progress/attempt been made towards this direction? Maybe restricting to some special class of groups (hyperbolic, relatively hyerpolic, nilpotent etc.)?
 A: As mentioned by Yves, the idea of constructing a quasi-isometry between two spaces from a "nice" homeomorphism between their "boundaries" applies to hyperbolic spaces (typically hyperbolic groups). It is a statement proved by Frédéric Paulin (Un groupe hyperbolique est déterminé par son bord, Journal of the London Mathematical Society, 54 (1), 50-74, 1996).
Theorem. Let $X$ and $Y$ be two geodesic and proper hyperbolic spaces admitting non-elementary isometry groups with bounded quotients. Any I-quasiconformal or quasi-Möbius homeomorphism $\partial X \to \partial Y$ is induced by a quasi-isometry $X \to Y$.
This theorem was generalised very recently by Ruth Charney, Matthew Cordes and Devin Murray to Morse boundaries (Quasi-Mobius homeomorphisms of Morse boundaries, arxiv:1801.05315). Namely:
Theorem. Let $X$ and $Y$ be two proper, cocompact, geodesic metric spaces. Assume that $\partial_*X$ contains at least three points. Then a homeomorphism $\partial_*X \to \partial_*Y$ is induced by a quasi-isometry $X \to Y$ if and only if it is $2$-stable and quasi-Möbius.
For hyperbolic spaces, the Morse boundary coincides with the usual boundary. So the latter statement holds for spaces whose Morse boundaries have cardinality at least three. (Interestingly, the class of finitely generated groups admitting a Morse boundary with at least three points remains mysterious. But acylindrically hyperbolic groups belong to this class, so we at least know that many such groups exist.)
However, I don't really know applications of these statements. 
