Applications of p-deficiency and p-largeness

TL;DR

This paper explores the properties of groups with high p-deficiency, demonstrating their largeness and Golod-Shafarevich characteristics, and extends known results to Coxeter groups for odd primes.

Contribution

It establishes new links between p-deficiency, p-largeness, and Golod-Shafarevich properties, and generalizes Grigorchuk's results to Coxeter groups for odd primes.

Findings

Groups with p-deficiency > 1 are large.

Such groups are Golod-Shafarevich for primes ≥ 7.

Extension of Grigorchuk's results to Coxeter groups for odd primes.

Abstract

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large, which implies that Schlage-Puchta's infinite finitely generated p-groups are not finitely presented. We also show that for all primes p at least 7, any group having a presentation of p-deficiency greater than 1 is Golod-Shafarevich, and has a finite index subgroup which is Golod-Shafarevich for the remaining primes. We also generalise a result of Grigorchuk on Coxeter groups to odd primes.