This should probably be an easy question, but I don't know how to answer it: Suppose *G* is a finitely generated presentable group. Suppose *a* is the absolute minimum of the sizes of all generating sets for *G* and *b* is the absolute minimum of the number of relations over all presentations of *G*. Question: Is it necessary that *G* has a presentation that simultaneously has *a* generators and *b* relations?


The case *b = 0* is just the fact that a free group cannot be generated by fewer elements than its free rank.


The problem could probably be interpreted in terms of CW-complexes (where the generators give rise to 1-cells and the relators give rise to 2-cells) but, because of my lack of familiarity with CW-complexes, I don't immediately see how to use these to solve the problem.


It also seems to be related to the notion of "deficiency" of a group, which is the (maximum possible over all presentations) difference #generators - #relations (under the opposite sign convention, the minimum possible difference #relations - #generators).



Does every finitely presentable group have a presentation that simultaneously minimizes the number of generators and number of relators?
A stronger question, is the deficiency of $G$ realized for a presentation with
the minimal number of generators (rank($G$))? This question is asked in a [paper of Rapaport](http://www.ams.org/mathscinet-getitem?mr=308277), and proved to be true for nilpotent and 1-relator groups. 


Addendum:


The question appears as Question 2, p. 2, of a [book by Gruenberg](http://books.google.com/books?id=1PQ1BVOl7WMC&lpg=PP7&ots=3Q_OnZqmeW&dq=gruenberg%2520relation%2520modules&lr&pg=PA2#v=onepage&q&f=false). Lubotzky
has answered the analogous question affirmatively in the category of profinite
groups ([Corollary 2.5](http://www.ams.org/mathscinet-getitem?mr=1848964)). (Note though that this is in the category of profinite presentations, so it does not imply an affirmative answer even for finite groups). 


