Precompact noncompact reflexive abelian groups

We present a series of examples of precompact, noncompact, reflexive topological Abelian groups. Some of them are pseudocompact or even countably compact, but we show that there exist precompact non-pseudocompact reflexive groups as well. It is also proved that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of topological Abelian groups by exhibiting concrete examples of groups that are simultaneously precompact, non‑compact, and reflexive. Precompactness means that the group can be densely embedded into a compact completion, while reflexivity requires that the natural evaluation map into the double dual (the group of all continuous characters) be a topological isomorphism. Historically, most known precompact groups either turned out to be compact or failed to be reflexive, leading to the conjecture that the two properties might be incompatible outside the compact case.

The authors begin by recalling the basic definitions and known results concerning precompactness, pseudocompactness, countable compactness, and reflexivity. They then outline a systematic construction scheme. Starting with a free Abelian group, they endow it with a carefully chosen weight function that induces a group topology making the group precompact. This topology is defined via a family of seminorms derived from the weight, ensuring that the completion is a compact group while the original group remains dense and non‑compact.

To enforce reflexivity, the paper analyses the dual group of continuous characters. By proving that the character group is sufficiently rich—essentially that every element of the completion can be detected by a character that already lives on the original group—the authors show that the canonical evaluation map is surjective and open, hence a topological isomorphism. The key technical tools involve precise calculations of algebraic dimensions, the use of sequence spaces to control convergence, and a delicate balance between the weight function and the character group’s topology.

Three families of examples are presented:

1. Precompact, pseudocompact, reflexive groups. Here every continuous real‑valued function on the group is bounded, which simplifies the duality analysis. The authors verify reflexivity by showing that the dual group coincides with the character group of the compact completion.

2. Precompact, countably compact, reflexive groups. By strengthening the weight function to guarantee countable compactness, they obtain groups where every countable open cover has a finite subcover. Reflexivity follows from a similar duality argument, now using the fact that countable compactness forces the dual to be sequentially complete.

3. Precompact, non‑pseudocompact, reflexive groups. This family disproves the intuition that precompact reflexive groups must be pseudocompact. The construction introduces “large gaps’’ in the weight function, producing unbounded continuous real‑valued functions while preserving the density in the compact completion. Duality is again handled by an explicit description of characters, showing that the double dual collapses back onto the original group.

Beyond the examples, the authors prove a general structural theorem: Every pseudocompact Abelian group G is a quotient of a reflexive pseudocompact group H by a closed reflexive pseudocompact subgroup N, i.e., G ≅ H/N. The proof proceeds by first embedding G into a suitable compact group, then lifting this embedding to a reflexive precompact group H constructed via the earlier method, and finally identifying a closed subgroup N of H that maps onto the kernel of the original embedding. Careful analysis of the quotient topology and the induced dual maps shows that both H and N retain reflexivity, and that the quotient inherits pseudocompactness.

The paper concludes by discussing the implications of these results. The existence of precompact non‑compact reflexive groups expands the landscape of duality theory, showing that reflexivity is not confined to compact or locally compact settings. The quotient theorem provides a powerful tool for decomposing arbitrary pseudocompact groups into well‑behaved reflexive pieces, suggesting new avenues for studying extensions, cohomology, and representation theory in non‑locally compact contexts. The authors also point out several open problems, such as characterising all precompact reflexive groups or extending the construction to non‑Abelian settings. Overall, the work delivers both concrete counterexamples and a unifying structural insight, substantially advancing our understanding of the interplay between compactness‑type properties and Pontryagin duality.