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I am trying to find an example of a discrete group of Möbius transformation that is isomorphic (algebraically) to a non-discrete group.

Can someone please help finding such groups.

hbghlyj
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yaa09d
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  • Think of translations along the same axis, one of length $\sqrt{2}$, the other of $\sqrt{3}$... –  Oct 03 '13 at 06:04
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    Consider the cyclic group generated by $z\mapsto z+1$ and the cyclic group generated by $z\mapsto e^{i}z$. – 23rd Oct 03 '13 at 06:06
  • Actually, now I'm seeing how you tagged this and wondering what you mean by Mobius transformations. –  Oct 03 '13 at 06:06
  • @SteveD I mean a subgroup of isometries of the upper-half plane (subgroup of PSL$(2, R)$) – yaa09d Oct 03 '13 at 06:19
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    Ahh, then the question is much more difficult. In fact, examples do exist, but they are not easy to come up with "on your own". Probably the easiest way to do this is to take two hyperbolic geodesics that are ultra-parallel, and then consider the translation isometry on each. These generate a rank 2 free subgroup, and Jorgensen's inequality can be used to find a non-discrete version. A "complete" solution to this problem is in a series of papers by Gilman and Maskit. Note that looking at 2-generator groups is natural because being discrete is detected on 2-generator subgroups. –  Oct 03 '13 at 06:44
  • If you mean subgroups of $PSL(2,\Bbb R)$, do you know how to modify the example in my last comment? – 23rd Oct 03 '13 at 07:40
  • I admit that I am confused by exactly what you want in this question, based on the tag "riemann-surfaces". It's late here; I'm going to bed. –  Oct 03 '13 at 07:49
  • @Landscape Yes. I know what you mean. Thank you. – yaa09d Oct 03 '13 at 16:04
  • @SteveD I added the tag of "riemann-surfaces" because compact Riemann surfaces can be written as the quotient of the upper-half plane by a discrete subgroup of $PSL(2,\Bbb R)$ . I just wanted to know if such disceret groups can be algebraically isomorphic to non-discrete groups. – yaa09d Oct 03 '13 at 16:11
  • @yaa09d: You are welcome. – 23rd Oct 03 '13 at 16:16
  • @yaa09d: Yes I thought you wanted a non-discrete subgroup which was isomorphic to a surface group. For a closed surface, I don't know any explicit examples of non-discrete representations, but they definitely exist by standard density arguments. Here is a nice MO discussion about the genus 2 case: http://mathoverflow.net/questions/100130/a-question-about-hyperbolic-double-torus –  Oct 03 '13 at 17:29

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Here is the simplest example: Take an infinite cyclic group $G_1$ generated by a hyperbolic (or parabolic if you prefer) isometry of the hyperbolic plane. This group is discrete. Now, take an infinite cyclic group $G_2$ generated by an irrational rotation (an elliptic isometry of infinite order). Clearly, the second group is not discrete as it is dense in a subgroup of $PSL(2,R)$ isomorphic to $S^1$. If you want to see more interesting examples, read Steve D's comments. Furthermore, there are nondiscrete subgroups of $PSL(2, R)$ isomorphic to fundamental groups of closed hyperbolic surfaces. Even more:

Every finitely generated discrete infinite subgroup $G$ of $PSL(2,R)$ is isomorphic to a nondiscrete subgroup, except for a finite number of Van Dyck groups (I can list the exceptions if you are interested).

Edit. Here is how to construct explicit examples of nondiscrete groups of surface group embeddings. Start for instance with the Coxeter triangle group $W$ whose Coxeter graph is the triangle with the edge labels $7$. This group acts as a cocompact discrete group of isometries of the hyperbolic plane with the fundamental domain which is an equilateral hyperbolic triangle $T$ with the angles $\pi/7$. Now, take an equilateral hyperbolic triangle $T'$ with the angles $2\pi/7$. It exists since $6\pi/7<\pi$. Now, take the group $W'$ generated by reflections in the edges of the new triangle. There exists an obvious homomorphism $f: W\to W'$ sending generators to generators. It requires a bit of work to show that $f$ is an isomorphism (surjectivity is clear, injectivity follows from Galois group considerations). Now, $W$ admits a finite index subgroup isomorphic to the fundamental group of closed hyperbolic surface (I can compute the genus, but it would take me some time). Restriction to this subgroup would be an explicit example, since vertices and edges of $T'$ one can compute explicitly using Lorentzian model of the hyperbolic plane.

Moishe Kohan
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  • Thank you for the detailed answer. Could you give me an example of a nondiscrete subgroup of $PSL(2,R)$ isomorphic to a fundamental groups of a closed hyperbolic surface, please? Also could you give me some references to this topic, please? Thank you. – yaa09d Oct 03 '13 at 16:18