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 $\langle C\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B| + |C| + 2$. 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 $\langle C\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B| + 5$. Moreover, if $\langle C\rangle$ is not isomorphic to the Klein bottle group, i.e., the group with the presentation $\langle x, y \;|\; xyx = y\rangle$, then for all subsets $B$ of G with $|B|\geq 5$, $|BC|\geq |B| + 5$. 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 $\{x\in G \;|\; \alpha_x \neq 0\}$. 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 $|supp(\beta)|\geq 12$. Also, we prove that if $\alpha \beta= 1$ for some $\alpha \beta \in F[G]$ such that $|supp(\alpha)| = 3$, then |supp(\alpha)|\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.