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

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.

• If you have constructed those 8 Mobius transformations for $\pi_1(\Sigma_2)$, isn't that the Fuchsian group of genus 2 you are looking for? (By the way, I would think it's only 4 transformations, but maybe you are counting e.g. $a$ and $a^{-1}$ separately.)
– Ted
Commented Dec 29, 2020 at 7:59
• The OP doesn't understand that there are 3 kind of genus 2 Riemann surfaces: those given as a compact quotient of $\Bbb{H}$, those given as a smooth projective curve, those given by an arbitrary atlas of charts (for example the compactification of the quotient of $\Bbb{H}$ by a congruence subgroup). Going from the latter to the former is the uniformization theorem. I don't know if the uniformization for projective curves is easier, in which space of 2-forms will we find the hyperbolic metric ? Commented Dec 29, 2020 at 19:36

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.

1. 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.