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

Abstract

Let [Formula: see text] be a unique product group, i.e. for any two finite subsets [Formula: see text] of [Formula: see text], there exists [Formula: see text] which can be uniquely expressed as a product of an element of [Formula: see text] and an element of [Formula: see text]. We prove that if [Formula: see text] is a finite subset of [Formula: see text] containing the identity element such that [Formula: see text] is not abelian, then, for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Also, we prove that if [Formula: see text] is a finite subset containing the identity element of a torsion-free group [Formula: see text] such that [Formula: see text] and [Formula: see text] is not abelian, then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Moreover, if [Formula: see text] is not isomorphic to the Klein bottle group, i.e. the group with the presentation [Formula: see text], then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. The support of an element [Formula: see text] in a group algebra [Formula: see text] ([Formula: see text] is any field), denoted by [Formula: see text], is the set [Formula: see text]. By the latter result, we prove that if [Formula: see text] for some nonzero [Formula: see text] such that [Formula: see text], then [Formula: see text]. Also, we prove that if [Formula: see text] for some [Formula: see text] such that [Formula: see text], then [Formula: see text]. 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.