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 $A$ and $B$ of $G$ there exists $x\in G$ 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 $|B|\ge 7$, $|BC|\ge |B|+|C|+2$. 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 $|B|\ge 7$, $|BC|\ge |B|+5$. 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 $|B|\ge 5$, $|BC|\ge |B|+5$. The support of an element $\alpha ={\sum }_{x\in G}{\alpha }_{x}x$ a group algebra $F[G]$ ($F$ is any field), denoted by $supp(\alpha )$, is the set $\{x\in G|{\alpha }_x\ne 0\}$. By the latter result, we prove that if $\alpha \beta =0$ for some non-zero $\alpha ,\beta \in F[G]$ such that $|supp(\alpha )|=3$, then $|supp(\beta )|\ge 12$. Also, we prove that if $\alpha \beta =1$ for some $\alpha \beta \in F[G]$ such that $|supp(\alpha )|=3$, then |supp(α)|\geq 10$. These results improve a part of results in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667-693] and Dykema et al. [Exp. Math., 24 (2015), 326-338] to arbitrary fields, respectively.