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

2023/03/14 by Fisher, Sam P., Sánchez-Peralta, Pablo
#12E15 #20F65 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2303.08165

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 kG be the division ring containing kG constructed by Lewin and Lewin. We prove that 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.

Related

• On Zero Divisors with Small Support in Group Rings of Torsion-Free Groups
• On group rings of virtually abelian groups
• Kaplansky's zero divisor and unit conjectures on elements with supports of size 3
• Group rings of three-manifold groups
• Zero divisor and unit elements with support of size 4 in group algebras of torsion free groups