Compact Lie groups isolated up to conjugacy

Abstract

The set $\mathcal{S}(G)$ of compact subgroups of a Hausdorff topological group $G$ can be equipped with the Vietoris topology. A compact subgroup $K\in \mathcal{S}(G)$ is isolated up to conjugacy if there is a neighborhood $\mathcal{U}\subseteq \mathcal{S}(G)$ of $K$ such that every $L\in \mathcal{U}$ is conjugate to $K$. In this paper, we characterize compact subgroups of a Lie group that are isolated up to conjugacy. Our characterization depends only on the intrinsic structure of $K$, the ambient Lie group $G$ and the embedding of $K$ into $G$ are irrelevant. In addition, we prove that any continuous homomorphism from a compact group $G$ onto a compact Lie group $H$ induces a continuous open map from $\mathcal{S}(G)$ onto $\mathcal{S}(H)$.