An algorithm for low dimensional group homology

TL;DR

This paper presents an algorithm leveraging Hopf's formula to estimate the second homology of finitely presented groups with finite field coefficients, demonstrated through examples involving special linear groups and related to Quillen's conjecture.

Contribution

It introduces a novel algorithm that computes low-dimensional group homology using Hopf's formula, with practical examples for specific groups.

Findings

Algorithm successfully estimates H_2(G;k) for given groups.

Examples include computations for SL_2 over rings of integers.

Results relate to and support aspects of Quillen's conjecture.

Abstract

Given a finitely presented group $G$, Hopf's formula expresses the second integral homology of $G$ in terms of generators and relators. We give an algorithm that exploits Hopf's formula to estimate $H_2(G;k)$, with coefficients in a finite field k, and give examples using $G=SL_2$ over specific rings of integers. These examples are related to a conjecture of Quillen.