Zero divisors and units with small supports in group algebras of torsion-free groups

Abstract

Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field &, the group ring &[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in &[G] whose supports have size 3. For any field & and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈&[G] such that |supp(α)| = 3, then |supp(β)|≥10. If & = &2 is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈&[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields. © 2018 Taylor & Francis.