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Let $G$ be a Fuchsian group with normalizer $N(G)$ inside $PSL(2,13)$

Due to the Hurwitz formula, it suffices to find a presentation of $G$ of the form: $$\langle x_1,\ldots,x_r,a_1,b_1,\ldots,a_\gamma,b_\gamma: x_1^{m_1} = \cdots = x_r^{m_r} = [a_1\;b_1]\cdots[a_\gamma\;b_\gamma]x_1\cdots x_r = 1\rangle$$

In that case, the genus (of the compactification of the riemann surface $G\backslash\mathcal{H}$) is given by $$2g - 2 = [N(G):G]\left(2\gamma - 2 + \sum_{i=1}^r\left(1-\frac{1}{m_i}\right)\right)$$

Thus, my question reduces to - Is there in general a nice way to find such a presentation of the group?

In particular, suppose $G$ is the Fuchsian group derived from the Hurwitz Quaternion Order $\mathcal{Q}_\text{Hur}$ (defined here: - namely $G$ is the subgroup of $\mathcal{Q}_\text{Hur}$ consisting of elements of norm 1 that are also 1 mod $\mathfrak{p}\mathcal{Q}_\text{Hur}$, where $\mathfrak{p}$ is one of the three primes lying above 13 in the ring of integers of $\mathbb{Q}(2\cos(2\pi/7)))$.

Is there a nice way to find the genus(or equiv. a presentation in terms of elliptic and hyperbolic elements) of this group?

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I think your group $G$ will be torsion-free, in which case one can compute the genus from Gauss-Bonnet and the order of the quotient group, which should be something like $PSL_2(13)$. See also: – Ian Agol Oct 1 '12 at 17:05

You are essentially asking for a fundamental domain for your Fuchsian group $G$. This is an area of specialty for John Voight, who has produced an algorithm for finding such a fundamental domain and thus a presentation for $G$:

In particular, he pays special attention to the case of quaternion algebras. I'm a little bit confused by a few things in your question: e.g., claiming that a Fuchsian group can have a normalizer inside of a finite group, but what I propose should work.

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See Exotic arithmetic structure on the first Hurwitz triplet, where this particular case was studied by Lei Yang.

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