Cardinality of product sets in torsion-free groups and applications in group algebras

Abstract

Let G be a unique product group, i.e. for any two finite subsets A,B of G, there exists x G which can be uniquely expressed as a product of an element of A and an element of B. We prove that if C is a finite subset of G containing the identity element such that (C) is not abelian, then, for all subsets B of G with |B|≥ 7, |BC|≥|B| + |C| + 2. Also, we prove that if C is a finite subset containing the identity element of a torsion-free group G such that |C| = 3 and (C) is not abelian, then for all subsets B of G with |B|≥ 7, |BC|≥|B| + 5. Moreover, if (C) is not isomorphic to the Klein bottle group, i.e. the group with the presentation (x,y|xyx = y), then for all subsets B of G with |B|≥ 5, |BC|≥|B| + 5. The support of an element α =x Gαxx in a group algebra [G] ( is any field), denoted by supp(α), is the set {x G|αx0}. By the latter result, we prove that if αβ = 0 for some nonzero α,β [G] such that |supp(α)| = 3, then |supp(β)|≥ 12. Also, we prove that if αβ = 1 for some α,β [G] such that |supp(α)| = 3, then |supp(β)|≥ 10. These results improve a part of results in Schweitzer [J. Group Theory 16(5) (2013) 667-693] and Dykema et al. [Exp. Math. 24 (2015) 326-338] to arbitrary fields, respectively. © 2020 World Scientific Publishing Company.