On 𝑑-independent locally compact groups

Abstract

Abstract A topological group is called 𝑑-independent if, for every subgroup of cardinality less than the continuum, there exists a countable dense subgroup intersecting it trivially. This notion was introduced by Márquez and Tkachenko, who characterized 𝑑-independent second-countable locally compact abelian groups in purely algebraic terms via 𝑀-groups, and asked whether the same characterization holds without metrizability assumptions. In this paper, we answer this question affirmatively. We prove that every separable locally compact abelian 𝑀-group is 𝑑-independent, thereby extending the result of Márquez and Tkachenko to the non-metrizable setting. We also investigate 𝑑-independence in the non-abelian context. In particular, we show that every separable connected compact group is 𝑑-independent. The proof reduces the problem to simple connected compact Lie groups and relies on the structure of maximal tori and the existence of dense free subgroups.