The Density Nucleus of a Topological Group

Given a topological groupG (usually compact abelian), the authors study the poset D = D(G) of dense subgroups of G and its impact on the algebraic structure of G. A key tool for this is the subgroup den(G) := ∩ D. Definition. For a cardinal κ ≥ 1, a topological group is in the class Ff (κ) [F(κ); F2(κ); Fad(κ), respectively] if some family of κ-many dense subgroups of G is independent and freely generated [independent; pairwise independent; pairwise almost disjoint, respectively]. Let K be a compact abelian group. Then 1. K ∈ Fad(2); 2. K ∈ F(2) ⇔ either r(K) > 0 or each leading Ulm-Kaplansky invariant of K is infinite; 3. there are D0, D1 ∈ D(K) such that den(K) = D0 ∩D1; 4. K ∈ F(κ) ⇔ K ∈ F2(κ); 5. if K is torsion and K ∈ F(2), then K ∈ F(κ) ⇔ κ ≤ each leading Ulm-Kaplansky invariant of K; 6. if r(K) > 0, then K ∈ F(κ) ⇔ κ ≤ r(K); if, in addition, r(K) ≥ d(K), then K ∈ Ff (κ) ⇔ K ∈ F(κ).