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).