Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then $\phi$ induces a homeomorphism between their boundaries.

The proof of the above statement is well-written in Bridson and Haefliger's book.

My question is that `can we drop the condition that $X$ and $Y$ are proper?'. In some papers about boundaries of hyperbolic spaces, the authors usually say that the above theorem is true without mentioning that $X$ and $Y$ are proper. If you know the answer or any references, then let me know.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.