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

TL;DR

This paper proves that for certain classes of groups, their group algebras can be embedded into division rings with specific properties, confirming longstanding conjectures and establishing new structural results.

Contribution

It demonstrates Hughes-free embeddings of group algebras for virtually compact special and 3-manifold groups, confirming Kaplansky's Zero Divisor Conjecture and related conjectures.

Findings

Kaplansky's Zero Divisor Conjecture holds for torsion-free 3-manifold groups.

Group algebras of virtually compact special groups are coherent.

Hughes-free embeddings are established for these classes of 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.