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!



Reference request: flat surfaces
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).


