Narain Gupta's three normal subgroup problem and group homology

TL;DR

This paper explores the role of third homology in solving Narain Gupta's three normal subgroup problem, providing a detailed description of certain subgroup intersections in group rings under specific homological conditions.

Contribution

It applies homological methods to classical group ring problems, specifically linking third homology to the structure of subgroup intersections in free groups.

Findings

Third homology is crucial in understanding subgroup intersections.

A complete description of $F igcap (1+{f rst})$ is provided under certain conditions.

The results depend on torsion properties of homology groups.

Abstract

This paper is about application of various homological methods to classical problems in the theory of group rings. It is shown that the third homology of groups plays a key role in Narain Gupta's three normal subgroup problem. For a free group $F$ and its normal subgroups $R,\,S,\,T,$ and the corresponding ideals in the integral group ring $\mathbb Z[F]$, ${\bf r}=(R-1)\mathbb Z[F],\ {\bf s}=(S-1)\mathbb Z[F],\ {\bf t}=(T-1)\mathbb Z[F],$ a complete description of the normal subgroup $F\cap (1+{\bf rst})$ is given, provided $R\subseteq T$ and the third and the fourth homology groups of $R/R\cap S$ are torsion groups.