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Sep 15, 2012 at 16:55 vote accept ARG
Sep 14, 2012 at 1:34 comment added Misha Gjergji: Yes, he does mention it, I forgot the reference though, I think it was unpublished at the time. I will add few examples where geometric boundary is preserved by quasi-isometries.
Sep 14, 2012 at 1:21 comment added Gjergji Zaimi I believe Lyons references an earlier paper where the first example of quasi isometric Riemannian manifolds with non-homeomorphic Martin boundaries was established. Also, I think your point 3 is interesting, and it would be cool to know of a list of conditions that guarantee a good behavior of boundaries under quasi isometries.
Sep 14, 2012 at 1:00 history answered Misha CC BY-SA 3.0