Groups of p-deficiency one

TL;DR

This paper investigates groups with p-deficiency one, showing they have finite index subgroups surjecting onto free groups and do not possess Kazhdan's property (T), extending previous results on p-large groups.

Contribution

It proves that groups with p-deficiency one have finite index subgroups surjecting onto free groups and lack Kazhdan's property (T), extending earlier work on p-large groups.

Findings

Groups with p-deficiency one have finite index subgroups surjecting onto free groups.

Such groups do not have Kazhdan's property (T).

The results extend previous theorems on p-large groups.

Abstract

The main result of [4] is that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto . This implies that these groups do not have Kazhdan's property (T). Additionally, we prove that the main result of [4] implies a result of Lackenby [8].