I am thinking of 3-manifolds as arising from Heegaard splittings which I am thinking about in terms of Heegaard diagrams. I know that 3-manifold groups are rather special in the class of all finitely presentable groups. 


I know how to go from a Heegaard diagram ($g$ red curves and $g$ blue curves each forming a cut system on a genus $g$ closed orientable surface) to a presentation for $\pi\_1$ - namely I have $g$-generators coming from the red handlebody $g$ relations coming from the disks attached along the blue curves. To find the relations I just orient all of the curves and the surface and for each blue curve I read off the word in the red curves (with the signs of the intersections determining the sign of the words). 


I would like to know some explicit balanced group presentations that I can not get in this way - and I would like to see why I can't embed the curves if possible. Of course $< a ,b | [a,b] , [a,b]^2>$ for example fits the bill, since $\mathbb{Z}^2$ is not the fundamental group of a compact 3-manifold. But I would like a more "curves on surfaces" type explanation. 


Ideally, I would like some explicit necessary and/or sufficient condition for a word or collection of words to be realizable by embedded curves as above. 


Thanks!



Obstructions to realizing a balanced presentation as a 3-manifold group
$\newcommand{\ZZ}{\mathbb{Z}}$For your first question: the [Baumslag-Solitar](https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group) group $BS = BS(2,3)$ is not residually finite, so cannot embed in a [three-manifold group](https://mathoverflow.net/questions/61661/residual-finiteness-of-fundamental-group-of-compact-3-manifold). Thus $BS \times \ZZ$ has a balanced presentation, but is not a three-manifold group. A nice two-generator example is $\ZZ/3\ZZ \times \ZZ$. (For both examples, my proofs rely on geometrization. But see HJRW's comments below.)


For your second question: As Ryan says, the program [Heegaard](https://www.math.uic.edu/t3m/) is our friend here. There is extensive [documentation](https://www.math.uic.edu/.../heegaard/Documentation/Heegaard_Documentation.pdf) explaining the algorithm. It operates on the Whitehead graph of the given presentation. The algorithm performs a sequence Whitehead moves until it either gives up or it makes the Whitehead graph planar. If there is now a good "pairing" of the vertices then the algorithm has produced a Heegaard splitting of the desired three-manifold. 


Finally: see page 47, item C.4 of the book [3-manifold groups](https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwiP68aZ697ZAhXpK8AKHY0tD0oQFggpMAA&url=http%3A%2F%2Fwww.uni-regensburg.de%2FFakultaeten%2Fnat_Fak_I%2Ffriedl%2Fpapers%2F3-manifold-groups-final-version-031115&usg=AOvVaw156nUdc5c4BYbV-I8sKCtI) for a list of interesting references.


