Division rings for group algebras of virtually compact special groups and $3$-manifold groups

Abstract

Let k be a division ring and let G be either a torsion-free virtually compact special group or a finitely generated 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 Zero Divisor 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 and Linton, 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 that this happens when G is virtually compact special.