Recall, that for two metric spaces $X$ and $Y$ the Lipschitz distance between $X$ and $Y$ is the infimum over all bi-Libschitz maps $f:X\to Y$ of $$\log(\max({\rm dil}(f),{\rm dil}(f^{-1}))).$$
Here $\rm dil$ is the dilatation. This distance can be equal to $\infty$.
I have a question concerning the Example on page 71 of Gromov's book "Metric structures on Riemannian and non Riemannian spaces."
Gromov says in the example that Lispchitz distance defines a metric on the moduli space of compact Riemann surfaces of curvature $-1$.
Question. How to prove that the topology given by Lipschitz distance is the standard one? In other words, suppose that there is a sequence of compact curvature $-1$ surfaces that is Cauchy in Lipschitz metric. Why does it converge to a compact surface of curvature $-1$?
