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.

  • $\begingroup$ 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.) $\endgroup$
    – Ted
    Commented Dec 29, 2020 at 7:59
  • $\begingroup$ 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 ? $\endgroup$
    – reuns
    Commented Dec 29, 2020 at 19:36

1 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.

  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.


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