Bicyclic units, Bass cyclic units and free groups

Abstract

Let $G$ be a finite group and $\mathbb{Z}G$ its integral group ring. We show that if $\alpha$ is a non-trivial bicyclic unit of $\mathbb{Z}G$, then there are bicyclic units $\beta$ and $\gamma$ of different types, such that $\GEN{α,β}$ and $\GEN{α,γ}$ are non-abelian free groups. In case that $G$ is non-abelian of order coprime with 6, then we prove the existence of a bicyclic unit $u$ and a Bass cyclic unit $v$ in $\mathbb{Z}G$, such that for every positive integer $m$ big enough, $\GEN{u^m,v}$ is a free non-abelian group.

15 pages