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I am quoting the following description from a paper,

"...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a genus $g$ Riemann surface is a subgroup of $PSL(2,\mathbb{C})$ that is freely generated by $g$ loxodromic elements of $PSL(2,\mathbb{C})$. Let the $g$ generators of the Schottky group be $\{ L_i \}_{i=1}^g$. Then Mobius transformations map circles to circles and in particular for these loxodromic transformations, $2g$ disjoint circles $\{ C_i, C'_i\}_{i=1}^g$ can be chosen such that $L_i(C_i) = C_i'$. Under the quotient $\mathbb{C}/\Gamma$ these circles in $\mathbb{C}$ map to $g$ nontrivial elements of the fundamental group. The circles generate a maximal freely generated subgroup of the fundamental group. The remaining $g$ generators of the fundamental group are then obtained by paths that connect the pairs of circles.."

  • Can someone explain how are "loxodromic elements of $PSL(2,\mathbb{C})$ defined? How are they to be constructed given a Riemann surface?

  • How are the circles $C_{i=1, ..,g}$ to be chosen?

  • What is the action $L_j (C_i)$?

  • How to see the paths between the $C_i$s as becoming generators of teh fundamental group?


Is there a review paper available (best if online!) where this construction is explained? I haven't seen this in Riemann surface books I know of!

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  • $\begingroup$ Did you try to google "Schottky groups", "loxodromic elements"? Read Ahlfors and Sario's book on Riemann surfaces? $\endgroup$ – Misha Mar 19 '14 at 3:04
  • $\begingroup$ If some loop in $\mathbb{C}/\Gamma$ lifts to $\mathbb{C}$ as a closed loop (those $C_i$), it is certainly nullhomotopic, so no nontrivial element in $\pi_1(\mathbb{C}/\Gamma)$. Are there maybe some points cut out of $\mathbb{C}$? $\endgroup$ – Achim Krause Mar 19 '14 at 3:05
  • $\begingroup$ @Misha I have seen other Riemann surface books - not this one - can you kindly link to some review reference from which I can quickly pick up this stuff? $\endgroup$ – user6818 Mar 19 '14 at 3:25
  • $\begingroup$ @AchimKrause Yes - in some sense - the Riemann surface being considered is an uniformization of a branched Riemann surface - the original branching is about a finite number of line-segments. $\endgroup$ – user6818 Mar 19 '14 at 3:28
  • $\begingroup$ Cannot you always in fact restrict yourself to the quotient of the (either unit disk or) half-plane by a torsion free discrete subgroup of $\textrm{PSL}(2,\mathbb R)$? (genus <2 is clear anyway :)) $\endgroup$ – მამუკა ჯიბლაძე Mar 19 '14 at 5:26
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A very clear explanation of uniformization by Schottky groups can be found in Ch. X of L. Ford's book, Automorphic functions, Mcgraw Hill, 1929.

The proof is not constructive. Riemann surface is a sphere with $g$ handles. Cut every handle, and you obtain a topological sphere with $2g$ holes. By a theorem of Koebe this is conformally equivalent to the Riemann sphere with $2g$ round holes. All known proofs of the Koebe theorem are highly non-constructive. Even in the simplest cases, it is a challenge to compute the generators of the group from an explicitly given Riemann surface. Ford's exposition has an advantage that it is very geometric and intuitive.

In the cite you give there is one incorrect sentence: the Riemann surface is not $C/\Gamma$. It is $\Omega/\Gamma$, where $\Omega$ is the discontinuity set of the Schottky group.

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  • $\begingroup$ @Alexandro Eremenko Any online lecture notes is available on this? And its not just the uniformization issue - what is this construction of C_i and loxodromic elements of PSL(2,C) ? How does this quotient of $\mathbb{C}$ by the Schottky group relate to the fundamental group of the Riemann surface? $\endgroup$ – user6818 Mar 19 '14 at 6:35
  • $\begingroup$ @user6818: Here (archive.org/details/introductiontoth00forduoft) is a preliminary version (1915) of Ford's book indicated by Alexandre. $\endgroup$ – ACL Mar 19 '14 at 9:39
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    $\begingroup$ @user6818: The construction of loxodromic elements is complicated, and in most cases not explicit. This IS the uniformization issue. Like with Fuchsian group uniformization. The relation between the surface and the group is always complicated. $\endgroup$ – Alexandre Eremenko Mar 19 '14 at 13:15
  • $\begingroup$ @AlexandreEremenko Thanks! So these circles $C_i$ are to be chosen on the uniformizing surface (and NOT on the Riemann surface) - right? How are these circles chosen? Does this choice somehow fix the $g$ loxodromic elements of $PSL(2,\mathbb{C})$ that need to be chosen? Can you define what are these "loxodromic" elements? And how/why does the fundamental group play into this? [...I am searching for the referenced books - but it would help to get some view of the basic before I penetrate the books!..] $\endgroup$ – user6818 Mar 19 '14 at 16:17
  • $\begingroup$ @AlexandreEremenko Can you define what you mean by the "discontinuity set"? Do you mean the fundamental domain of the action of the Schottky group on the uniformizing surface? $\endgroup$ – user6818 Mar 19 '14 at 16:18
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I would suggest reading Vicki Chuckrow's very nice 1968 paper, and references therein.

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  • $\begingroup$ Since this concept is not the central focus of my research I am a bit skeptic of having to delve into huge details about it - I hope to get a functional understanding of it - so that I can do this construction algorithmically as and when I need it for the various Riemann surfaces I face. I guess this paper will be a faster way than the Ford's book? $\endgroup$ – user6818 Mar 20 '14 at 8:34
  • $\begingroup$ Also more fancifully is there any analogue of this in 3-manifolds? Like they can also be gotten as some such general quotient construction? $\endgroup$ – user6818 Mar 20 '14 at 8:35
  • $\begingroup$ Knowing about you, you would know anything about 3-manifolds! :D $\endgroup$ – user6818 Mar 20 '14 at 8:43

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