Area of fundamental domain of Fuchsian group and index of a Fuchsian group in the triangle group

Question

Let $\mathbb{H}$ be the upper half plane model of hyperbolic geometry. Let $\Gamma$ be the Fuchsian group such that $\mathbb{H}/\Gamma$ is the compact orientable surface of genus $2$.

Suppose $\Gamma = \langle g_1,g_2,g_3,g_4 \mid g_1g_2g_1^{-1}g_2^{-1}g_3g_4g_3^{-1}g_4^{-1}=1 \rangle$ where $g_1,g_2,g_3,g_4$ are hyperbolic translations.

In B. Maskit's book on Kleinian Groups, it is given that the normal form of a hyperbolic translation is given by the matrix $$\frac{1}{x-y}\begin{pmatrix}xk^{-1}-yk&xy(k-k^{-1})\\ k^{-1}-k & xk-yk^{-1} \end{pmatrix}$$ where $x$ and $y$ are fixed points of the translation and $k^2$ is the multiplier.

Then we have four 3 tuples $(x_i,y_i,k_i)$ for $i=1,2,3,4$ for our $g_i$'s. Now the fundamental domain of $\Gamma$ will be a hyperbolic octagon.

The area of the fundamental domain will be $4\pi$.

Can we determine some identity or equation in terms of the parameters $x_i,y_i, k_i$ and $4\pi$?

Is it possible to find the index of $\Gamma$ in the $(2,3,7)$ triangle group in terms of the parameters $x_i,y_i, k_i$?

Answer 1

EDIT: My answer (below) is for the original question. The current question has been modified to include my answer.

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The area of a fundamental domain for $\Gamma$ is $4\pi = -2\pi \chi(\mathbb{H} / \Gamma)$. There are various proofs; most people think of this as a consequence of the Gauss-Bonnet theorem.

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To answer your next question:

Is it possible to find the index of $\Gamma$ in the $(2,3,7)$ triangle group in terms of the parameters?

When $\Gamma$ is a subgroup at all (which happens only for very very very special values of the parameters) it has index $84$. This is because the area of the $(2,3,7)$ orbifold is $\frac{\pi}{21}$.