Peter Loth

One—dimensional compact connected abelian groups and nonsplitting extensions.

‎‎‎The Pontryagin dual of any (discrete) abelian group written as a factor group of a free abelian group can be described as a closed subgroup of a torus. Using this approach, the one—dimensional compact connected abelian groups, called solenoids, are classified and constructed as topological subgroups of the torus \mathbb{T}^{\aleph_0}. For any solenoid \Sigma\neq\mathbb{T}, we exhibit a nonsplitting extension of \Sigma by a profinite group, dual to a nonsplitting extension 0\to {\rm tor}(A)\to A\to F\to 0 of abelian groups where F is a rank–1 torsion–free group \neq \mathbb{Z}. The constructed groups A are generalizations of Examples of Fuchs and are simply presented, i.e., they can be defined by generators and relations that involve at most two generators.

‎‎‎Much of this talk is based on joint work with Dikran Dikranjan, Wayne Lewis and Adolf Mader.