# Fundamental group of a closed hyperbolic surface is Gromov hyperbolic

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

Thanks!

## migrated from mathoverflow.netOct 10 '14 at 21:51

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• 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. – quid Oct 10 '14 at 21:24
• This question is not research-level - any introductory text will contain a proof of this fact. Voting to close. – HJRW Oct 10 '14 at 21:30

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.