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A special case of Theorem 10.6.4 of Beardon's book about possible shapes of Dirichlet domains have been discussed in the following result from Yiltekin-Karatas's dissertation.

Lemma 1 (Page.9)
Suppose that $ F $ is a fundamental domain for $ \Gamma $. Then let $ l_1, l_2, l_3 $ be the lengths of the cycles that correspond to the conjugacy classes associated with $ m_1, m_2 $, and $ m_3 $. Also, suppose that there are $ s $ accidental cycles of lengths $ r_1, \ldots, r_s $. We choose any $ p $ in $ F^\circ $ and connect $ p $ to the vertices of $ F $. By the Gauss-Bonnet formula, the area of $ F $ equals: $ \mu(F) = (l_1 + l_2 + l_3 + r_1 + \ldots + r_s)\pi - 2\pi - 2\pi s - 2\pi \left( \frac{1}{m_1} + \frac{1}{m_2} + \frac{1}{m_3} \right) $

This formula relates the area of the fundamental domain $ F $ to the lengths of its cycles and accidental cycles, as well as the associated conjugacy classes and their lengths. Note that Theorem 10.6.4 of Beardon's book has the same proof but it doesn't mention that this relation has been derived from Gaussi-Bonnet theorem for hyperbolic triangles.

Here is the Gauss-Bonnet theorem;

Theorem 8 (Gauss-Bonnet)
Let $ \Delta $ be a hyperbolic triangle with angles $ \alpha, \beta, \gamma $. Then the area of the triangle is: $ \mu(\Delta) = \pi - \alpha - \beta - \gamma $

QUESTION :
Can someone provide a derivation to explain the connection between the formula for $\mu(F)$ in Lemma 1 and the Gauss-Bonnet theorem or other relevant results?

Extra

I could find a similar expression from Beardon but still it has some differences, e.g., it doesn't contain the term for a number of accidental cycles. But still there seems to be a gap to derive the target expression above.

Theorem 7.15.1
If $P$ is any polygon with interior angles $\theta_1, \dots, \theta_n$, then $\text{h-area}(P) = (n - 2) \pi + (\theta_1 + \cdots + \theta_n).$

Reference

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    $\begingroup$ I got one very useful & related reference www2.math.upenn.edu/~shiydong/Math501X-7-GaussBonnet.pdf though I think , in essence , the Issue is figuring out a Decomposition in terms of triangles & then adding up the terms & simplifying that , to get the Conclusion. It is grunt work going through various articles & making up the connections. It would have been great if Beardon had given a Proof with the notation & terminology of that textbook. Indeed , the mismatching terminology between the references makes things unnecessarily complicated , @LeeMosher , which Beardon could have avoided. $\endgroup$
    – Prem
    Commented Apr 9 at 13:23
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    $\begingroup$ Theorem 7.15.1 [......] is derived from theorem 8 [......]. $\endgroup$
    – Lee Mosher
    Commented Apr 9 at 13:35
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    $\begingroup$ It [meaning Theorem 7.15.1] can be applied to Lemma 1 [...]. $\endgroup$
    – Lee Mosher
    Commented Apr 9 at 13:36
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    $\begingroup$ Presumably you mentioned Theorem 7.15.1 as a possible tool, and my comments are intended to confirm that. $\endgroup$
    – Lee Mosher
    Commented Apr 9 at 13:37
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    $\begingroup$ @Prem Thank you for providing the reference! hmm,, I understood that Gauss-Bonnet is a fundamental result from Differential geometry but I'm not quite faimilar with the domain.. So, I think I have to study DG before diving into your reference. I think I'll make another post w/h my current angle to the derivation of interest. $\endgroup$
    – Rowing0914
    Commented Apr 9 at 13:55

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