Extremal pseudocompact topological groups

Topological groups here are assumed to satisfy the Hausdorff separation property. A topological group G is totally bounded if it embeds as a (dense) subgroup of a compact group image; here image, the Weil completion of G, is unique in the obvious sense. It is known that every pseudocompact topological group is totally bounded; and a totally bounded group G is pseudocompact if and only if G meets each nonempty Gδ-subset of image. A pseudocompact group is said to be r-extremal [resp., s-extremal] if G admits no strictly finer pseudocompact group topology [resp., G has no proper dense pseudocompact subgroup]. (Note: r- derives from “refinementʹʹ, s- from “subgroupʹʹ.) Let image be the class of non-metrizable, pseudocompact Abelian groups. The authors contribute to the growing literature (see for example J. Galindo, Sci. Math. Japonicae 55 (2001) 627) supporting the conjecture that no image is either r- nor s-extremal—but that conjecture remains open. Except for portions of (a), the following are new results concerning image proved here. The proofs derive largely from basic, sometimes subtle, considerations comparing the algebraic structure of image with the algebraic structure of the Weil completion image. (a) If G is either r- or s-extremal, then image. (b) If G is totally disconnected, then G is neither r- nor s-extremal. (c) If G is either torsion-free or countably compact, then G is not both r- and s-extremal. (d) Not every closed, pseudocompact subgroup N of G is s-extremal; if G itself is either r- or s-extremal then the witnessing N may be chosen connected. (e) If in some image every closed subgroup in image is r-extremal, then there is connected image with the same property. (f) If 2ω=2ω1 and every closed subgroup of G is pseudocompact, then G is not s-extremal.