Stably free modules over virtually free groups

TL;DR

This paper proves that for certain group rings involving free groups and finite nilpotent groups, there are infinitely many stably free modules of rank 1, revealing new algebraic structures.

Contribution

It demonstrates the existence of infinitely many stably free modules over group rings formed from free groups and finite nilpotent groups of non square-free order.

Findings

Infinitely many stably free modules of rank 1 exist over ${f Z}[G\times F_m]$.

Results apply to groups with non square-free order.

Extends understanding of projective modules over group rings.

Abstract

Let $F_m$ be the free group on $m$ generators and let $G$ be a finite nilpotent group of non square-free order; we show that for each $m\ge 2$ the integral group ring ${\bf Z}[G\times F_m]$ has infinitely many stably free modules of rank 1.