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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 $x_i,y_i, k_i$?

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  • $\begingroup$ It may help us (to help you) if you say a few words about where the questions come from. Is this self study? A course? A graduate writing project? $\endgroup$
    – Sam Nead
    Aug 25, 2022 at 15:25
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    $\begingroup$ @SamNead This is a self-study. I am studying $[3^7]$ tiling on genus two surfaces and want to know how many distinct tilings are possible via studying their fundamenatal domains. Here we can say that two fundamental domains are equivalent if the tiling on the surface are isomorphic. Then I want to know number of non equivalent fundamental domains. $\endgroup$
    – KAK
    Aug 26, 2022 at 2:16

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


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.


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 $\frac{\pi}{21}$.

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  • $\begingroup$ Thank you for your answer. Yes, that is true but I need it in terms of the parameters and want to equate it with $4\pi$. $\endgroup$
    – KAK
    Aug 25, 2022 at 11:02
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    $\begingroup$ @KAK : the point is that it doesn’t depend on the parameters. $\endgroup$
    – HJRW
    Aug 25, 2022 at 11:14
  • $\begingroup$ @KAK - Are you looking for some kind of (integral?) identity? Perhaps you could edit your post to ask your actual question... $\endgroup$
    – Sam Nead
    Aug 25, 2022 at 11:24
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    $\begingroup$ Integrate $d\mathrm{Area}$ over the fundamental domain. We can (locally) arrange matters so that the bounds on the domain are (very unpleasant) functions of the parameters. $\endgroup$
    – Sam Nead
    Aug 25, 2022 at 13:08
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    $\begingroup$ The "limits of integration" is exactly the boundary of the fundamental domain. Since the domain is (very) non-unique, to "find" the boundary one makes many choices and then does a lot of hyperbolic geometry. It is all pretty convoluted. I have only done a small number of particularly nice cases (making many mistakes along the way). Finding fundamental domains for the once-punctured torus is much easier, but still requires work. However, I am not going to try to do that in a comment box on mathoverflow... $\endgroup$
    – Sam Nead
    Aug 25, 2022 at 15:00

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