An algorithm for low dimensional group homology

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.

💡 Research Summary

The paper addresses the long‑standing computational difficulty of determining the second homology group of a finitely presented group when coefficients are taken in a finite field. The classical Hopf formula, dating back to the 1930s, expresses (H_{2}(G;\mathbb Z)) as the quotient ((R\cap