The weight of a countably compact group whose cardinality has countable cofinality

Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Square of countably compact groups without non-trivial convergent sequences

We describe the structure of totally disconnected minimal ω \omega - bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p p -adic integers Z p \mathbb {Z}_{p} . In this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo. As an application we answer the question (posed in Pseudocompact and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775–790) when arbitrary powers of minimal ω \omega -bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals.

There exists an infinite family of p p -compact groups whose Weyl groups correspond to the finite p p -adic pseudoreflection groups G ( q , r , n ) G(q,r,n) of family 2a in the Clark-Ewing list. In this paper we study these p p -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical J J -homomorphism. Finally, we also describe a faithful complexification homomorphism from these p p -compact groups to the p p -completion of unitary compact Lie groups.