Deficiency of a presentation
The deficiency of a presentation of a group is the number of generators minus the number of relations in a presentation.
(Note that in some conventions, it is defined as the number of relations minus the number of generators. All the deficiency values would thus get negated and all occurrences of minimum would get replaced by maximum and vice versa).
Note that since trivial and redundant relations can be thrown into any presentation, the deficiency of a presentation can be pushed to as low as we please. The challenge is in obtaining a presentation with high deficiency. The deficiency of a group is the maximum possible deficiency over all presentations of a group.