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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, and proved to be true for nilpotent and 1-relator groups.

The question appears as Question 2, p. 2, of a book by Gruenberg. Lubotzky has answered the analogous question affirmatively in the category of profinite groups (Corollary 2.5). (Note though that this is in the category of profinite presentations, so it does not imply an affirmative answer even for finite groups).

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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, and proved to be true for nilpotent and 1-relator groups. Also

The question appears as Question 2, p. 2, of a book by Gruenberg. Lubotzky has answered the analogous question affirmatively in the category of finitely presented pro-$p$ profinite groups , the answer is yes according to Lemma 1.1 of a paper of Lubotzky(Corollary 2.5). Lubotzky also remarks (Note though that this is in the analogue category of Lemma 1.1 is unlikely to hold profinite presentations, so it does not imply an affirmative answer even for discrete groups. It would make sense to first try to find such a groupfinite groups).
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, and proved for nilpotent and 1-relator groups. Also, in the category of finitely presented pro-$p$ groups, the answer is yes according to Lemma 1.1 of a paper of Lubotzky. Lubotzky also remarks that the analogue of Lemma 1.1 is unlikely to hold for discrete groups. It would make sense to first try to find such a group.