Fuchsian groups from explicit representations corresponding to genus $2$ surfaces

Question

I am learning hyperbolic geometry on my own. I have worked out the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation, that is, as follows:

Let $\Sigma_2$ is a genus $2$ Riemann surface. It can be thought of in the following way: it is the quotient space $\mathbb{H}/\pi_1(\Sigma_2)$ where an element of $\pi_1(\Sigma_2)$ can be thought of as a biholomorphic map of upper half plane $\mathbb{H}$. So, for an explicit representation of $\pi_1(\Sigma_2)$, the fundamental group of the genus two surface, into $PSL(2,\mathbb {C})$, I have constructed $8$ Mobius transformations (mainly, I have constructed $4$ Mobius transformations, also I am considering their inverses) and those satisfy the defining relation of fundamental group of genus $2$ surface $\pi_1(\Sigma_2)$.

$\textbf{Question:}$ Can I get Fuchsian group of genus $2$ surface from this? Or, can I get Fuchsian group of genus $g$ surface in general?

Also, I have heard that the explicit representation of $\pi_1(\Sigma_2)$, fundamental group of the genus two surface, into $PSL(2,\mathbb{C})$ so that the upper half plane, modulo the induced action of $\pi_{1}$, is a "$\textbf{hyperbolic structure}$" on the topological surface $\Sigma_{2}$. And this is somehow related to theory of Tiechmuller Space. I want to learn Tiechmuller theory. Kindly advise a road-map to get this.

Please help me. Thanking in advanced.

Answer 1

1. For references to the Teichmuller theory, see for instance, here, both the accepted answer and comments. However, prior to this, I suggest reading something more introductory, such as Katok's book "Fuchsian Groups" or/and

Anderson, James W., Hyperbolic geometry, Springer Undergraduate Mathematics Series. London: Springer. ix, 230 p. (1999). ZBL0934.51012.

2. If all what you have is merely four elements $a_1, b_1, a_2, b_2$ of $PSL(2, {\mathbb R})$ satisfying the surface group relator $$ [a_1,b_1][a_2,b_2]=1, $$ this is far from enough for defining a Fuchsian group (whose quotient space will be a genus 2 surface $\Sigma_2$).

The missing conditions are:

i. $a_1, b_1, a_2, b_2$ generate a discrete subgroup $\Gamma$ of $PSL(2, {\mathbb R})$.

ii. $\Gamma$ is isomorphic to $\pi_1(\Sigma_2)$. Equivalently, $[a_1,b_1][a_2,b_2]=1$ is a defining relator of the group $\Gamma$ (informally, all other relators are its consequences).

In case you are wondering, the constructions described in the answers to your MO question here satisfy the missing conditions.