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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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    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

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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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    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