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?
Best Answer
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 $\Gamma$.