Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model defined?
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1$\begingroup$ How do you define a distance function on a Riemannian manifold? In your case, the quotient is an orbifold, but did you try to use the same formula? $\endgroup$ – Misha Sep 21 '15 at 19:38

2$\begingroup$ What is "the Fuchsian model". Almost certainly not research level, voting to close. $\endgroup$ – Igor Rivin Sep 21 '15 at 20:00