Classifying Spaces of Subgroups of Profinite Groups

TL;DR

This paper studies the topological space formed by all closed subgroups of a profinite group, analyzing its classification and complexity through properties like homeomorphism and scattered height.

Contribution

It provides new classifications and bounds on the complexity of the space of closed subgroups of profinite groups.

Findings

Classification of the space of subgroups up to homeomorphism

Establishment of bounds on the scattered height of the space

Insights into the topological complexity of subgroup spaces

Abstract

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.