Answer to a question of Ros\l{}anowski and Shelah

TL;DR

This paper proves that in any locally compact non-discrete group, there exists a subgroup that is null but non-meager, affirming a conjecture about the structure of such groups.

Contribution

It extends previous results by confirming the existence of null but non-meager subgroups in all locally compact non-discrete groups, not just specific cases.

Findings

Existence of null but non-meager subgroups in all locally compact non-discrete groups

Affirmative answer to Ros extl{}anowski and Shelah's question in the general case

Generalization beyond the Cantor group and reals

Abstract

In [arXiv:1605.02261] Ros\l{}anowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with $ZFC$ that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper we give affirmative answers for the general case.