Generators for Group Homology and a Vanishing Conjecture

TL;DR

This paper develops methods to find generators of the second homology group of finitely presented groups over finite fields, relating to a conjecture of Quillen, with example calculations illustrating the approach.

Contribution

It introduces a new technique for generating the second homology of finitely presented groups over finite fields, expanding on previous work and addressing a conjecture of Quillen.

Findings

Provides explicit generators for H_2(G;F_p)

Connects the problem to Quillen's conjecture

Includes example calculations demonstrating the method

Abstract

Letting $G=F/R$ be a finitely-presented group, Hopf's formula expresses the second integral homology of $G$ in terms of $F$ and $R$. Expanding on previous work, we explain how to find generators of $H_2(G;\mathbb{F}_p)$. The context of the problem, which is related to a conjecture of Quillen, is presented, as well as example calculations.