# Gromov's boundary at infinity, drop the hypothesis on hyperbolicity

It's an easy result that if we have two quasi isometric hyperbolic spaces, then their Gromov boundaries at infinity are homeomorphic.

I found online these notes where at page 8, prop 2.20 they seem to drop the hypothesis on hyperbolicity. They give two references (french articles) for the proof but I didn't found anything.

Hence, can someone give me a reference/proof/counterexample to the following statement?

Let X,Y proper geodesic spaces, and let $f\colon X \to Y$ be a quasi isometry between them, then $\partial_pX \cong \partial_{f(p)}Y$

• This is not exactly the answer you are looking for but it looks like boundary in those notes is only defined for $\delta$-hyperbolic so I am guessing that statement implicitly has the spaces hyperbolic. – Paul Plummer Mar 13 '15 at 1:34
• that was I thinking actually. Because in the two references I just spot the word hyperbolic in the statements of the proposition :( Anyway I hope someone can provide a counterexample, I'm curious now :) Thanks for the comment! – Riccardo Mar 13 '15 at 7:42
• Which definition of boundary do you use? I think they are not all equivalent for non-hyperbolic spaces. – Seirios Mar 13 '15 at 8:26
• @Seirios I'm used to "working" (actually studying) with the boundary as the set of geodesic rays passing through a point p in which we identify two geodesics iff their distance is bounded by a costant – Riccardo Mar 13 '15 at 9:12