Which infinite abelian groups admit an almost maximally almost-periodic group topology?

A topological group G is said to be almost maximally almost-periodic if its von Neumann radical is non-trivial, but finite. In this paper, we prove that every abelian group with an infinite torsion subgroup admits a (Hausdorff) almost maximally almost-periodic group topology. Some open problems are also formulated.

💡 Research Summary

The paper investigates the existence of Hausdorff group topologies on infinite abelian groups for which the von Neumann radical (the kernel of the Bohr compactification) is non‑trivial but finite; such groups are called “almost maximally almost‑periodic”. The von Neumann radical n(G) can be described as the intersection of kernels of all continuous characters, and a group is maximally almost‑periodic when n(G)=1 and minimally almost‑periodic when n(G)=G. Earlier work showed that every infinite abelian group admits a non‑maximally almost‑periodic topology, but it left open whether the radical can be forced to be finite.

The main contributions are two theorems. Theorem A asserts that any abelian group possessing an infinite torsion subgroup admits a Hausdorff topology with a non‑trivial, finite von Neumann radical. Theorem B is more precise: for any prime p and any non‑zero element x in the Prüfer p‑group ℤ(p^∞), there exists a Hausdorff group topology τ on ℤ(p^∞) such that n(ℤ(p^∞),τ)=⟨x⟩. The case p≠2 had already been settled by Lukács; the novelty of this paper lies in handling the remaining case p=2.

The proof strategy proceeds in several stages. First, a structural lemma (Lemma 2.5) shows that any infinite abelian group contains a subgroup isomorphic to ℤ, a Prüfer group ℤ(p^∞), or an infinite direct sum of non‑trivial finite cyclic groups. For the direct‑sum case, Lukács’ earlier result (Theorem 2.6) already provides an almost maximally almost‑periodic topology. Thus the essential difficulty reduces to constructing suitable topologies on Prüfer groups.

The authors employ the machinery of T‑sequences, originally introduced by Zelenyuk and Protasov. A sequence {a_n} in a group G is a T‑sequence if there exists a Hausdorff group topology τ on G making a_n→e. The finest such topology is denoted G{a_n}. The Zelenyuk‑Protasov criterion (Theorem 2.2) characterises T‑sequences via combinatorial conditions on finite linear combinations of the a_n’s. In almost torsion‑free groups (including Prüfer groups) a simplified condition (Theorem 2.3) applies.

For p≠2, Lukács used a “canonical form” for elements of ℤ(p^∞) that expresses each element uniquely as a finite integer combination of the standard generators e_n with coefficients constrained to a small set. This representation enables the construction of a T‑sequence whose associated topology has the desired radical. However, the same canonical form fails for p=2 because of the different arithmetic of 2‑powers.

To overcome this, the paper introduces a new canonical form for ℤ(2^∞) (Definition 3.2). Every element y can be written uniquely as y=∑σ_{2n}e_{2n} with coefficients σ_{2n}∈{−1,0,1,2} and only finitely many non‑zero. Theorem 3.3 proves existence, uniqueness, and a minimality property of this form using a specially designed function f(x)=max{−2x,x}. Lemmas 3.4–3.9 develop technical estimates: they bound the sum of f(σ_{2n}), control the support size λ(y), and relate the order of differences y−z to the positions of non‑zero coefficients. These estimates guarantee that the sequences constructed later satisfy the Zelenyuk‑Protasov condition, thus are T‑sequences.

With the canonical form in hand, the authors construct, for a given non‑zero x∈ℤ(2^∞), a sequence {a_k} whose terms have rapidly increasing orders (ensuring the condition n_{k+1}−n_k→∞). Lemma 2.4 then confirms that {a_k} is a T‑sequence. The finest Hausdorff topology making a_k→e has von Neumann radical exactly ⟨x⟩, establishing Theorem B for p=2. Combining this with Lukács’ result for odd primes yields Theorem B in full generality.

Finally, Theorem A follows: any infinite abelian group with an infinite torsion subgroup contains either a Prüfer subgroup (handled by Theorem B) or an infinite direct sum of finite cyclic groups (handled by Theorem 2.6). Hence such groups admit an almost maximally almost‑periodic Hausdorff topology.

The paper concludes with two open problems: (1) whether infinite torsion‑free abelian groups can admit almost maximally almost‑periodic topologies, and (2) what additional topological properties (metrizability, completeness, etc.) such topologies might possess. These questions point toward a deeper classification of abelian groups based on the size and structure of their von Neumann radicals.

Overall, the work extends the landscape of topological group theory by showing that the von Neumann radical can be forced to be any prescribed finite subgroup of a Prüfer group, and by resolving the previously missing case p=2 through a novel canonical representation and careful combinatorial analysis.