Infinite-fold strong topological minimal self-joinings for infinite abelian groups

Determine whether the abelian groups \(\bigoplus_{i\in\mathbb N}\mathbb Z/2\mathbb Z\) and \(\bigoplus_{i\in\mathbb N}\mathbb Z\) admit actions with infinite-fold strong topological minimal self-joinings.

Background

The paper studies restrictions on the order of strong topological minimal self-joinings for finitely generated abelian groups and constructs examples of groups admitting actions with strong topological minimal self-joinings of every finite order. The authors note that an abelian group admitting an infinite-fold strong topological minimal self-joining cannot be finitely generated.

It remains unresolved whether either of the two specific infinitely generated abelian groups ⨁i∈NZ/2Z\bigoplus_{i\in\mathbb N}\mathbb Z/2\mathbb Z or ⨁i∈NZ\bigoplus_{i\in\mathbb N}\mathbb Z admits an action with infinite-fold strong topological minimal self-joinings.

References

In particular, we do not know if $\bigoplus_{i\in N}Z/2Z$ or $\bigoplus_{i\in N}Z$ have $\infty$-fold sTMSJ.

— Non-determinism in group actions and topological minimal self-joinings  (2609.26479 - Bitar et al., 22 Sep 2026) in Section “Groups admitting infinite-fold sTMSJs”