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I am learning hyperbolic geometry on my own. I want to learn Fuchsian groups.

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$, $\pi_1(\Sigma_g)$ denotes the fundamental group of $\Sigma_g$. Now $\mathbb{H}/\pi_1(\Sigma_g)$ can be thought as $\Sigma_g$ where an element of $\pi_1(\Sigma_g)$ can be thought of as a biholomorphism of upper half plane $\mathbb{H}$. That is equivalent to finding the explicit description of genus $g$ surface as the upper half plane modulo group of Deck Transformation.

My question: Can I study (or get) Fuschian group from the construction? Is Fuchsian group related to it?

I want to study Fuchian group. Please help me.

Thanking in advanced.

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Fuchian groups are (by definition) discrete subgroups of PSL($2$, $\mathbb{R}$), and this is isomorphic to Aut($\mathbb{H}$), the group of biholomorphic maps from $\mathbb{H}$ to $\mathbb{H}$. As you mentioned, there is a connection between subgroups of automorphisms that act properly discontinuously induce a quotient that is a covering map $\mathbb{H} \rightarrow \Sigma_g$.

One of the requirements for a properly discontinuous action is that the subgroup of Aut($\mathbb{H}$) acting must be a discrete subgroup, and hence you get the Fuschian groups. Now, the Riemann surfaces that are covered by the half-plane are called hyperbolic surfaces, because you can induce the metric using the covering map.

This gives you a relation between Hyperbolic Riemann Surfaces and Fuschian Groups.

The book "Riemann Surfaces", by Hershell M. Farkas and Irwan Kra study some relations about these groups and the corresponding Riemann Surface in the chapter IV, specially IV.5. This may be a good starting point.

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  • $\begingroup$ Do you know how to make clear for which (closed) hyperbolic polygons $P\subset \Bbb{H}$ there are finitely many Möbius transformations sending one edge to the other such that when used as transition charts they give a compact Riemann surface structure on $P$ and those Möbius transformations generate a discrete subgroup $G$ of $PSL(2,\Bbb{R})$ such that $P= G\backslash \Bbb{H}$ ? $\endgroup$
    – reuns
    Commented Dec 21, 2020 at 19:17
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    $\begingroup$ I haven't study these in a very long time, but there's the Poincaré Polygon Theorem, that relates these two notions. A good reference for this discussion is John H. Hubbard's book Teichmüller Theory And Applications To Geometry, Topology, And Dynamics, Theorem 3.9.5. $\endgroup$ Commented Dec 21, 2020 at 19:30
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    $\begingroup$ Another cool reference for this is math.la.asu.edu/~paupert/CriderPoincarePolygonTheorem.pdf. The proof of this theorem actually gives you some methods of finding the generators of the Fuschian Group associated with the polygon. $\endgroup$ Commented Dec 21, 2020 at 19:36
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    $\begingroup$ Great, I would never have found those $2\pi / m_\mathcal{E}$ 'proper elliptic cycles' myself $\endgroup$
    – reuns
    Commented Dec 21, 2020 at 19:48
  • $\begingroup$ @user454229: If you have detailed followup questions like in your last comment, it is much better to put the question into a new post, rather than to bury it in a comment to an answer to an old post where very few users of this site will see it. $\endgroup$
    – Lee Mosher
    Commented Dec 22, 2020 at 15:09

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