  It is an immediate consequence of the results by Yiftach and Schlage-Puchta
that a presentation with p-deficiency greater than one defines a group with
positive rank gradient. By results of Button and Thillaisundaram, a finite
presentation with p-deficiency greater than one defines a p-large group. Both
Yiftach/Schlage-Puchta and Thillaisundaram extended these results to the
p-deficiency one case. In this paper we consider the case when the presentation
has p-deficiency less than or equal to one.
