New Examples of Torsion-Free Non-unique Product Groups

TL;DR

This paper introduces an infinite family of torsion-free groups that lack the unique product property, demonstrating complex algebraic structures with large sets lacking unique element representations.

Contribution

It provides new examples of torsion-free groups without the unique product property, expanding understanding of algebraic group structures.

Findings

Existence of infinite torsion-free groups without the unique product property

Construction of arbitrarily large sets with no uniquely represented elements

Demonstration of complex algebraic behaviors in these groups

Abstract

We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.