Modular distribution of topo-symmetric invariants

Establish the uniform distribution modulo gcd(|G|,|H|) of the topo-symmetric dimension among extensions in Ext_ts(G,H) for a finite group G and a discrete abelian group H; specifically, show that the number of topo-symmetric extensions E with dim(E) ≡ k (mod gcd(|G|,|H|)) is asymptotically |Ext_ts(G,H)|/gcd(|G|,|H|).

Background

The paper introduces invariants for topo-symmetric extensions, including a topo-symmetric dimension and a density measure. In the section on conjectures, the authors posit a uniform modular distribution for these invariants, exemplified by an explicit asymptotic equidistribution statement for the dimension modulo gcd(|G|,|H|).

A following remark indicates that while numerical evidence supports this uniformity for small cyclic groups, a general proof has not yet been established, underscoring the open status of the problem.

References

The invariants of topo-symmetric extensions, such as dimension and density, are distributed uniformly modulo \gcd(|G|,|H|): Numerical experiments for small cyclic groups suggest that this uniformity holds for most classes of extensions, but a general proof is still open.

— The Theory of Topo-Symmetric Extensions of Topological Groups (2510.00018 - En-naoui, 21 Sep 2025) in Conjectures and Open Problems, Subsection Modular Distribution (Conjecture and subsequent Remark)