Fundamental groups of certain 3-manifolds 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:


*

*$H^2\times R$


*universal cover of $SL_2(R)$


*$H^3$

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.
 A: 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
$$ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=x^2, a^x=a, b^x=b, c^x=c, d^x=d\rangle.$$
