Fundamental group of a closed hyperbolic surface is Gromov hyperbolic

Question

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

Thanks!

Answer 1

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.

Answer 2

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}
}