TITLE: Obstructions to realizing a balanced presentation as a 3-manifold group
QUESTION [8 upvotes]: 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!

REPLY [3 votes]: $\newcommand{\ZZ}{\mathbb{Z}}$For your first question: the Baumslag-Solitar group $BS = BS(2,3)$ is not residually finite, so cannot embed in a three-manifold group.  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 is our friend here. There is extensive documentation 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 for a list of interesting references.