2
$\begingroup$

Does anyone have a reference for the proof of the result in the title?

Thanks!

$\endgroup$

migrated from mathoverflow.net Oct 10 '14 at 21:51

This question came from our site for professional mathematicians.

  • $\begingroup$ Welcome to the site! If you would write a somewhat less terse version of the question you might get a better reception. You can still edit the question. $\endgroup$ – quid Oct 10 '14 at 21:24
  • $\begingroup$ This question is not research-level - any introductory text will contain a proof of this fact. Voting to close. $\endgroup$ – HJRW Oct 10 '14 at 21:30
2
$\begingroup$

More generally, let $X$ be a $n$-dimensional closed hyperbolic manifold and let $G$ denote its fundamental group. It is a standard theorem in Riemannian geometry that the universal cover of such a manifold is isometric to the hyperbolic space $\mathbb{H}^n$. Therefore, $G$ acts geometrically on $\mathbb{H}^n$ and we deduce from Milnor-Svarc lemma that $G$ and $\mathbb{H}^n$ are quasi-isometric. Of course, $\mathbb{H}^n$ is Gromov-hyperbolic and we conclude that $G$ so is.

$\endgroup$
4
$\begingroup$

Yes, the reference is Jim Cannon's article in Bedford, Keane, Series

@book{bedford1991ergodic,
  title={Ergodic theory, symbolic dynamics, and hyperbolic spaces},
  author={Bedford, Tim and Michael (Michael S.) Keane and Series, Caroline},
  year={1991},
  publisher={Oxford University Press}
}
$\endgroup$

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.