Selective survey on spaces of closed subgroups of topological groups

TL;DR

This paper surveys various topologies on the set of all closed subgroups of a topological group and explores their applications across multiple mathematical fields.

Contribution

It provides a comprehensive overview of topologizations of subgroup spaces and highlights their relevance in diverse areas like model theory and geometric group theory.

Findings

Different topologies on subgroup spaces are systematically compared.

Applications in topological groups, model theory, and geometric group theory are demonstrated.

The survey identifies key open problems and future directions in the study of subgroup topologies.

Abstract

We survey different topologizations of the set $\mathcal{S}(G)$ of all closed subgroups of a topological group $G$ and demonstrate some applications in Topological Grous, Model Theory, Geometric Group Theory, Topological Dynamics.