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

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 $\mathrm{\mid }B\mathrm{\mid }\ge 7$, $|BC|\geq |B| + |C| + 2$\mathrm{.}Also,weprovethatif$C$ is a finite subset containing the identity element of a torsion-free group $G$ such that $\mathrm{\mid }C\mathrm{\mid }=3$ and $⟨C⟩$ is not abelian, then for all subsets $B$ of $G$ with $\mathrm{\mid }B\mathrm{\mid }\ge 7$, $|BC|\geq |B|

• 5$\mathrm{.}Moreover,if$\langle C\rangle$ is not isomorphic to the Klein bottle group, i.e., the group with the presentation $\langle x, y ;|; xyx = y\rangle$,thenforallsubsets$B$ofGwith$|B|\geq 5$,$|BC|\geq |B| + 5$. 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 ${x\in G ;|; \alpha_x \neq 0}$\mathrm{.}Bythelatterresult,weprovethatif$\alpha \beta = 0$forsomenon-zero$\alpha,\beta \in F[G]$suchthat$|supp(\alpha)| = 3$, then $\mathrm{\mid }supp(\beta )\mathrm{\mid }\ge 12$. Also, we prove that if $\alpha \beta =1$ for some $\alpha \beta \in F[G]$ such that $\mathrm{\mid }supp(\alpha )\mathrm{\mid }=3$, 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.