An algorithm for low dimensional group homology

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 H2(G;k), with coefficients in a finite field k, and give examples using G=SL2 over specific rings of integers. These examples are related to a conjecture of Quillen.