Residually finite groups that do not virtually have the unique product property

TL;DR

This paper constructs a finitely generated residually finite group where every finite index subgroup contains a specific subgroup, demonstrating it does not have the unique product property at any finite index level.

Contribution

It provides a novel example of a residually finite group lacking the finite index subgroup with the unique product property, challenging previous assumptions.

Findings

Constructed a finitely generated residually finite group with specific subgroup properties

Showed that all finite index subgroups contain Promislow's group

Demonstrated the absence of the unique product property in finite index subgroups

Abstract

We construct a finitely generated residually finite group $G$ with the property that every finite index subgroup of $G$ contains a subgroup isomorphic to Promislow's group. Hence $G$ does not have a finite index subgroup with the unique product property.