Division rings for group algebras of virtually compact special groups and $3$-manifold groups
Sam P. Fisher, Pablo Sánchez-Peralta
TL;DR
This work tackles Kaplansky's Zero Divisor Conjecture in positive characteristic by constructing embeddings of crossed product algebras $k*G$ into division rings for torsion-free groups $G$ that are either locally indicable and virtually compact special or finitely generated torsion-free $3$-manifold groups. The authors develop and apply the notions of Hughes-free and Linnell division rings, together with a graph-of-rings framework, to obtain embeddings, establish uniqueness properties, and derive agrarian invariants that generalize $L^2$-Betti numbers. Key contributions include: (i) embedding results for $k*G$ in the targeted classes, with Hughes-free embeddings when $G$ is locally indicable; (ii) the Kaplansky Zero Divisor Conjecture for torsion-free $3$-manifold groups; (iii) a confirmation of Kielak–Linton’s conjecture in this broader (agrarian) setting; (iv) coherence results for certain one-relator groups via the graph-of-rings approach; and (v) a Hughes-free identification of Lewin–Lewin division rings with the corresponding $ obreak k*G$-division rings in characteristic zero. These results expand embedding techniques beyond characteristic zero, connect geometric group theory with noncommutative algebra, and provide robust tools for agrarian invariants and the study of zero divisors in group algebras.
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.