Coideals as remainders of groups distinguishing between combinatorial covering properties

TL;DR

This paper constructs specific subgroups of the Cantor space with Menger remainders that do not possess stronger covering properties, addressing open questions in the area of combinatorial topology.

Contribution

It provides the first consistent examples of subgroups with Menger remainders lacking stronger properties, answering open questions in the field.

Findings

Existence of subgroups with Menger remainders but no stronger properties

Counterexamples to previous conjectures about covering properties

Resolution of open questions posed by Bella, Tokgoz, and Zdomskyy

Abstract

In this paper we construct consistent examples of subgroups of $2^\omega$ with Menger remainders which fail to have other stronger combinatorial covering properties. This answers several open questions asked by Bella, Tokgoz and Zdomskyy (Arch. Math. Logic 55 (2016), 767-784).