Closed subgroups of compact groups having open Chabauty spaces

Abstract

Given a locally compact group G, we denote by ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)$ the set of closed subgroups of G equipped with the Chabauty topology, which is a compact Hausdorff. For a closed subgroup H of G, the Chabauty topology on ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(H)$ is equivalent to the subspace topology on $\{L\in {\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)\, \mid \, L\subseteq H\}$ inherited from ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)$, and so ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(H)$ becomes a closed subset of ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)$. In some cases, the subspace ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(H)$ is open in ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)$, but it can also fail to be open as the example of the trivial subgroup $\{0\}$ of the additive group $\mathbb {Z}$ shows. In this paper, we are interested to determine the subset $\mathcal {T}\!\left( G\right)$ of closed subgroups H of a compact group G such that ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(H)$ is open in ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)$. An explicit description of $\mathcal {T}\!\left( G\right)$ is given. The paper also contains some topological properties of $\mathcal {T}\!\left( G\right)$. As an application, we show that if G is a compact group and if the lattice ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(G)$ is isomorphic to the lattice ${\mathcal {S}\mathcal {U}\!\mathcal {B}}(\mathbb {Z}_p)$ of the group $\mathbb {Z}_p$ of p-adic integers, then G is topologically isomorphic to $\mathbb {Z}_l$, for some prime l.