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I'm starting my master's thesis on geometry/topology & group theory.

I'd like to know examples of fundamental groups of 3-manifolds having geometric structure of the following types:

  1. $H^2\times R$

  2. universal cover of $SL_2(R)$

  3. $H^3$

My first idea was to trace down the fundamental groups of the manifolds given as examples in Wikipedia, but for almost all of them I couldn't find a group presentation. Anyway, I think I should begin with the simplest examples...

Thank you for helping!

Edit. For the moment I'm more interested in torsion-free groups.

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    Pick a hyperbolic knot and —after learning how to construct the Wirtinger presentation— find the Wirtinger presentation of the fundamental group of its complement.2012-01-09
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    I don't think your question is focused enough... You could try explaining why you want these examples, and so on, so that people know what you are really looking for and provide more useful (to you) answers. (By the way, I have no idea about what you mean by your parenthetical remark «(affective, cognitive, epistemological»; is it a joke?)2012-01-09
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    Right, I know that knots provide lots of examples for 3). Mariano, I was actually looking for the big-picture (so, any kind of examples - no jokes) but I agree with you. I'll think how to improve the question. Thanks!2012-01-09
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    For $H^2 \times R$, you can simply use the product of a hyperbolic surface with a circle. The fundamental group of the result is the direct product of the fundamental group of the surface with a cyclic group. For $H^3$, it ought to be possible to track down a presentation for the fundamental group of Seifert–Weber space. For $SL_2(R)$, the easiest example is probably the unit tangent bundle for a hyperbolic surface.2012-01-09
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    Jim, thanks! The last two examples you give me are in Wikipedia. Unfortunately I still don't know the group presentations. Do you suggest any reference?2012-01-09
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    Add motivation. For all we know, you are an experimental psychologist running an experiment on how easy it to get a handful of mathematicians to give you a list of wierd objects...2012-01-09
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    Mariano, I don't know how to put it. I found the subject very very interesting but I'm actually a beginner on all this! I read about geometrization conjecture and the 8 models of geometry. I thought I would more easily understand the other 5 because so I didn't ask about them. Sorry...2012-01-09
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    You should probably specify if you want to do this for closed manifolds or not, since for example, $\mathbb{R}^3$ admits all these geometries. For a compact case, the complement of the trefoil knot admits both $\tilde{SL_2(\mathbb{R})}$ and $\mathbb{H}^2\times\mathbb{R}$ geometries.2012-01-09
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    it seems to me like you are looking at fundamental groups of universal covers of some manifolds in this case the fundamental group is trivial since the universal covers are simply connected2012-01-09

1 Answers 1

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In the closed case, the easiest way to do this is to construct surface bundles over a circle. For example, if I let my surface $S$ be a hyperbolic surface (say a 2-torus), then the fundamental group of my surface bundle will be an HNN extension of $\pi_1(S)\cong\langle a,b,c,d\ |\ [a,b][c,d]\rangle$. If I let the generator of $\pi_1(S^1)$ act periodically, I'll get a Seifert-fibred space, and otherwise, I'll get a hyperbolic space. Hyperbolic space admits an $\mathbb{H}^3$ geometry. The Seifert-fibred space admits either an $\mathbb{H}^2\times\mathbb{R}$ or $\widetilde{SL_2(\mathbb{R})}$ geometry, depending on whether it is (virtually) a trivial circle bundle over some surface or not. [That is, if it's Euler number is zero or not.]

Now to get the $\mathbb{H}^3$ geometry, simply pick a non-periodic (outer) automorphism of $\pi_1(S)$. So you can get a 3-manifold $M$ with the following fundamental group, for example: $$ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=1, a^x=ab, b^x=b, c^x=c^b, d^x=d^b\rangle.$$

To get the $\mathbb{H}^2\times\mathbb{R}$ geometry, just take the product $S\times S^1$; this gives a 3-manifold $M$ with fundamental group $$ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=1, a^x=a, b^x=b, c^x=c, d^x=d\rangle.$$

For the $\widetilde{SL_2(\mathbb{R})}$ geometry, I don't know how to give a nice HNN presentation. But since it is really just like the $\mathbb{H}^2\times\mathbb{R}$ case, but with non-zero Euler number, I can simply add a singular fibre above to get a 3-manifold $M$ with presentation $$ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=x^2, a^x=a, b^x=b, c^x=c, d^x=d\rangle.$$

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    The second sentence might be slightly misleading: The term "2-torus" often refers to the genus 1 surface (as a special case of the term $n$-torus) but what you're referring to, of course, is the genus 2 surface.2012-04-08
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    In fact, maybe I'm getting your answer wrong, but: The case where the action of the generator of $\pi_1(S^1)$ is periodic should correspond to mapping tori of finite order diffeos, right? In that case, the Euler number will always be zero. Also, a mapping torus of a hyperbolic surface is hyperbolic iff the monodromy is pseudo-Anosov (Farb-Margulit Thm. 13.4), so it seems you've left out the ones of reducible monodromy (please correct me if I'm wrong). Those, we've discussed a bit in an [earlier question](http://math.stackexchange.com/questions/57576/which-mapping-tori-are-seifert-manifolds).2012-04-09
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    @SørenFugledeJørgensen: It was asked simply for examples, not a list of all types of hyperbolic 3-manifolds, etc. The Euler number of a mapping torus with periodic automorphism is zero, but as you can see by the last example, I am including a singular fibre, so it is no longer zero. I ignored the reducible case because it can be effectively ignored thanks to JSJ decomp.2012-04-10
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    Your hyperbolic example is wrong: $b^x = b$ is not allowed in a hyperbolic three-manifold. More generally, "non-periodic" is not enough to ensure hyperbolicity. You also need to rule out the "reducible" case.2014-12-21