Division rings for group algebras of virtually compact special groupsLet $k * G$ be a crossed product of a division ring $k$ and a torsion-free
virtually compact special group $G$. We embed $k * G$ into a division ring
$\mathcal D$ and use this to confirm a recent conjecture of Kielak and Linton.
In the case where $G$ is locally indicable, we prove that $\mathcal D$ is
Hughes-free.
If $H$ is a torsion-free one-relator group, let $\overline{kH}$ be the
division ring containing $kH$ constructed by Lewin and Lewin. We prove that
$\overline{kH}$ is Hughes-free whenever a Hughes-free $kH$-division ring
exists. This is always the case when $k$ is of characteristic zero; in positive
characteristic, our previous result implies this happens when $H$ is virtually
compact special.
arxiv.org