Positive rank gradient and p-largeness for groups defined by presentations with p-deficiency less than or equal to one

TL;DR

This paper investigates groups defined by presentations with p-deficiency less than or equal to one, extending known results about positive rank gradient and p-largeness to this less-studied case.

Contribution

It extends the understanding of group properties like rank gradient and p-largeness to presentations with p-deficiency at most one.

Findings

Groups with p-deficiency ≤ 1 can have positive rank gradient.

Such groups can also be p-large under certain conditions.

The results generalize previous findings for higher p-deficiency cases.

Abstract

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.