ONE-DIMENSIONAL COMPACT CONNECTED ABELIAN GROUPS AND NONSPLITTING EXTENSIONS

The one-dimensional compact connected abelian groups, called solenoids, are classified and constructed as topological subgroups of the torus 𝕋ℵ0. For an arbitrary solenoid Σ≠𝕋, we exhibit a nonsplitting extension of Σ by a profinite group, dual to a nonsplitting extension 0→tor(A)→A→F→0 of abelian groups where F is a rank-1 torsion-free group ≠ℤ. The constructed groups A are generalizations of examples of Fuchs.