On weak Sierpiński sets in groups and free subgroups

Abstract

In this paper we discuss the problem of existence of so called weak Sierpiński sets in groups. It is known that group $G$ has a Sierpiński subset if and only if it contains a free subgroup. In their paper, Tomkowicz and Wagon conjectured that an analogous result holds also for the weaker condition. We derive a number of properties of groups with weak Sierpiński subsets and use them to prove the above mentioned conjecture. This is an improved version of arXiv:1805.11486. Some of the included proofs have been simplified and shortened.

The results elaborated in this paper have been improved and complemented in another work arXiv:1903.07043