  Let $k$ be a division ring and let $G$ be either a torsion-free virtually
compact special group or a torsion-free $3$-manifold group. We embed the group
algebra $kG$ in a division ring and prove that the embedding is Hughes-free
whenever $G$ is locally indicable. In particular, we prove that Kaplansky's
zerodivisor conjecture holds for all group algebras of torsion-free
$3$-manifold groups. The embedding is also used to confirm a conjecture of
Kielak and Linton. Thanks to the work of Jaikin-Zapirain, another consequence
of the embedding is that $kG$ is coherent whenever $G$ is a virtually compact
special one-relator group.
  If $G$ is a torsion-free one-relator group, let $\overline{kG}$ be the
division ring containing $kG$ constructed by Lewin and Lewin. We prove that
$\overline{kG}$ is Hughes-free whenever a Hughes-free $kG$-division ring
exists. This is always the case when $k$ is of characteristic zero; in positive
characteristic, our previous result implies this happens when $G$ is virtually
compact special.
