Classifying Spaces of Subgroups of Profinite Groups

The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered height.

💡 Research Summary

The paper investigates the natural profinite topology on the set Sub(G) of all closed subgroups of a profinite group G. By endowing Sub(G) with the topology generated by basic open sets of the form U_N = {H ≤ G | HN = G}, where N runs over the open normal subgroups of G, the authors obtain a compact, totally disconnected, Hausdorff space that is automatically 0‑dimensional (its clopen sets form a basis). The central aim is to classify these topological spaces up to homeomorphism for broad families of profinite groups and to measure their complexity using the notion of scattered height, a well‑known ordinal invariant for scattered spaces.

The first major result concerns free profinite groups F̂_d (d ≥ 2). The authors prove that Sub(F̂_d) is a scattered space whose scattered height is exactly ω, the first infinite ordinal. The proof proceeds by constructing a decreasing transfinite sequence of closed subgroups whose derived sets eventually become empty after ω steps. Each step corresponds to fixing a finite index normal subgroup and examining the induced subspace, showing that the process cannot terminate earlier than ω and cannot exceed ω because the space is compact.

Next, the paper treats finite p‑groups and more generally pro‑p groups of finite rank. For a pro‑p group G of rank n, the authors establish that Sub(G) is scattered with height n. The argument hinges on the structure of the lower central series and the fact that the centre Z(G) is non‑trivial in any non‑trivial finite p‑group. By iteratively factoring out the centre, one obtains a filtration whose associated subspaces each contribute one level to the scattered hierarchy. Consequently, the scattered height coincides with the minimal number of generators (or equivalently the dimension as a Z_p‑module) of G.

The paper then moves to arbitrary direct products G = ∏{i∈I} G_i of profinite groups. It shows that Sub(G) is homeomorphic to the product of the spaces Sub(G_i) equipped with the product topology. As a corollary, the scattered height satisfies  ht(Sub(G)) = sup{i∈I} ht(Sub(G_i)). If the index set I is infinite, the supremum is always ≤ ω, so infinite direct products have scattered height at most ω. This result clarifies how the topological complexity of a product is governed by its most complex factor.

A significant portion of the work is devoted to bounding the scattered height from above and below in terms of algebraic invariants. For hereditarily just‑infinite profinite groups (those whose every non‑trivial closed normal subgroup has finite index), Sub(G) is shown to be non‑scattered; thus its scattered height is undefined. Conversely, for countably generated abelian profinite groups, Sub(G) is scattered and its height equals the Prüfer rank of G. The authors also treat mixed cases, such as groups with a non‑trivial central series, where the height equals the length of the central series.

The authors conclude by discussing applications. In Galois theory, the absolute Galois group of a field is a profinite group, and Sub(G) encodes the lattice of intermediate field extensions; understanding its topology can shed light on the “shape” of the field’s algebraic closure. In the theory of automorphism groups of profinite structures, the scattered height provides a new invariant that distinguishes otherwise algebraically similar groups. Finally, the classification results suggest a roadmap for a full homeomorphism classification of Sub(G) for all profinite groups, analogous to the classical classification of compact totally disconnected spaces via Cantor–Bendixson analysis.

Overall, the paper establishes a clear bridge between algebraic properties of profinite groups and the topological structure of their subgroup spaces, offering precise classifications, tight bounds on scattered height, and a suite of tools for future investigations.