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When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic:

  1. The universal cover $\widetilde{\Sigma}$ equipped with the pullback of any flat metric with conic singularity is quasi-isometric to the hyperbolic plane.
  2. Define two flat metrics $g_1$ and $g_2$ by means of holomorphic $n$-differentials $U_1$ and $U_2$. Then the pullbacks of $g_1$ and $g_2$ to $\widetilde{\Sigma}$ are quasi-isometric and the ratio of quasi-isometry can be controlled by $\|U_1-U_2\|$.

Please either give me a reference or a simple proof/explanation. Any help is appreciated!

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    $\begingroup$ (1) is probably not what you mean, since all compact metric spaces are quasi-isometric to each other. Do you mean the pullback metric on the universal cover? or do you mean quasi-conformal instead of quasi-isometric? $\endgroup$
    – John Pardon
    Commented Mar 13, 2015 at 20:11
  • $\begingroup$ You are right, I actually work on the universal cover. I've edited the question. $\endgroup$
    – Xin Nie
    Commented Mar 13, 2015 at 20:34

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For the first fact, a good reference is Jim Cannon's survey in Bedford/Keane/Series (you use fact the universal cover is quasi-isometric to the fundamental group, which is proved at great length by Cannon).

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