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For the regular 1-holed torus, we can study it by applying periodicity conditions on the plane

$$(x,y)\sim(x+2\pi,y)$$ $$(x,y)\sim(x,y+2\pi)$$

I saw that we can represent a $g$-holed torus with a polyhedron with $4g$-sides, which are oppositely identified. See for example question 1 and question 2.

The problem is that couldn't find a way to write periodicity conditions, like we did for the 1-holed torus. Is there such a way to impose periodicity conditions on the plane so that we can recover the topology of a $g$-holed torus.

Maybe the question can be rephrased as : Is there a quotient of the complex plane that gives us the $g$-holed torus, and if so, what are the exact equations that identify the points?

EDIT: The comment section suggested that a quotient of the hyperbolic plane should be used rather than the complex plane, to obtain periodicity conditions for the $g$-torus. If that is the case, how should I go about to find those periodicity conditions?

ghost
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    If I am understanding your question correctly then the answer is "no": a $g$-holed torus is the quotient of hyperbolic space, and cannot be the quotient of the complex plane. This is due to curvature arguments (which are hinted at in the second linked question, with the discussion of Euler characteristic). – user1729 Jun 07 '21 at 10:02
  • @user1729 OK. If its the quotient of the hyperbolic space, what are the periodicity conditions? – ghost Jun 07 '21 at 10:39
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    I only wrote a comment because I wasn't confident I could answer the whole question in complete depth :-) But if you want to know periodicity conditions, you first need to define the model of the hyperbolic plane you are working with. – user1729 Jun 07 '21 at 10:43
  • @user1729 No problem, your comment was quite helpful actually. But I don't have a model of the hyperbolic plane that I am currently working with. I just want to find a similar periodicity condition for the $g$-holed torus. Any guide for further reading is well appreciated my friend :) – ghost Jun 07 '21 at 11:02

1 Answers1

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As mentioned by @user1729 in the comments, the genus $g$ surface $\Sigma_g$ is a quotient of the hyperbolic plane $\mathbb{H}$ by some Fuchsian group $\pi_1 \Sigma_g \cong \Gamma \subset \operatorname{Isom}(\mathbb{H})$. Listing "periodicity conditions" is tantamount to finding explicit generators of $\Gamma$: for each generator $\gamma$, the corresponding periodicity condition is $\gamma \cdot x \sim x$ for $x \in \mathbb{H}$.

Finding these generators is just some hyperbolic trigonometry: see this answer for more details.

If you just want a final answer, then Wikipedia claims (but I have not checked) that the following works. Take the upper half plane model of $\mathbb{H}$, so that $\operatorname{Isom}(\mathbb{H})$ is given by the group $PSL_2(\mathbb{R})$ of linear fractional transformations. For $0 \leq k < 2g$, define $$a_k = \begin{pmatrix} \cos k\alpha & -\sin k\alpha \\ \sin k\alpha & \cos k\alpha \end{pmatrix} \begin{pmatrix} e^p & 0 \\ 0 & e^{-p} \end{pmatrix} \begin{pmatrix} \cos k\alpha & \sin k\alpha \\ -\sin k\alpha & \cos k\alpha \end{pmatrix},$$ where $\alpha = \frac{\pi(2g-1)}{4g}$, $\beta = \frac{\pi}{4g}$, and $p = \ln \frac{\cos \beta + \sqrt{\cos 2\beta}}{\sin \beta}$.

Then $\{a_0, a_1, \ldots, a_{2g-1}\}$ is a set of generators for $\Gamma$.

JHF
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