Edgar Márquez Rodríguez

D-independent topological groups.

A topological group G with |G|>1 is called d-independent if for every subgroup S of G with |S|<2^\omega, one can find a countable dense subgroup H of G such that S\cap H=\{e\}. Therefore, d-independent groups are separable and have cardinality at least 2^\omega. In this talk, we will discuss some properties of d-independent topological groups. For example, we present a characterization of separable metrizable d-independent abelian groups and show that products of separable topological groups can often be d-independent, even if the factors fail to be d-independent. Our main result is a purely algebraic characterization of d-independence in the class of compact metrizable abelian groups. We prove that a compact metrizable abelian group G with |G|>1 is d-independent if and only if for every integer m\geq 1, either |mG|= 2^\omega or |mG|=1. This characterization implies that a compact metrizable abelian group is d-independent if and only if it is maximally fragmentable [Comfort and Dikranjan, Topology Proc. 44 (2014), 325–356] iff G is an M-group as defined by D. Dikranjan and D. Shakhmatov in [Advances Math. 286 (2016), 286–307].