# Hausdorff distance in compact Lie groups

Let $$G$$ be a compact Lie group with a compatible biinvariant metric $$d$$. The hyperspace $$K(G)$$ of nonempty compact subsets of $$G$$ is a compact metric space with the Hausdorff metric, and it is easy to check that subgroups of $$G$$ form a closed subspace in $$K(G)$$, hence we may talk about the (compact) space of closed subgroups of $$G$$. Let us denote this space by $$\mathbf{K}(G)$$.

General question:

(1) Does anyone know any source that may help exploring spaces of the form $$\mathbf{K}(G)$$?

I have a conjecture:

(2) For a compact connected Lie group $$G$$ the following are equivalent:

a) $$G$$ is a limit point in $$\mathbf{K}(G)$$ (that is, it can be approximated by proper closed subgroups).

b) The circle group is a quotient of $$G$$.

Is it true? ( b)$$\implies$$a) is easy, take inverse images of finite subgroups of the circle group by the quotient map.)

For (1) I have found only the papers of Fischer and Gartside: On the space of subgroups of a compact group I and On the space of subgroups of a compact group II.

They mostly deal with arbitrary compact $$G$$ or profinite $$G$$, not Lie groups.

For (2) I found the MO question Approximating Lie groups by finite groups, which says that only compact abelian Lie groups can be approximated by finite subgroups (it refers to a paper of A. M. Turing, Finite Approximations to Lie Groups).

• These questions can be tricky; one can generalize this to a decent topology on the space of closed subgroups of any topological group, called the Chabauty space. Chabauty space on $\mathbb{R}^2$ is a $4$-sphere, but the topology of the chabauty space of $\mathbb{R}^n$ is not precisely known for $n>2$. There are works by e.g. Haettel and de la Harpe-Kleptsyn-de Cornulier on this. Apr 25, 2022 at 13:50
• Somebody commented here in April but later they deleted the comment. It contained extremely useful information for us: a paper of Mongomery and Zippin Together with Nicolas Tholozan's answer it helped us in our work with the space of closed subgroups.
– chj
Jun 10, 2022 at 15:03

Assume a sequence of subgroups $$G_n$$ converges to $$G$$. Up to extraction, we can assume that $$\mathrm{Lie}(G_n)$$ converges to a Lie subalgebra $$\mathrm{Lie}(H)$$. Since the adjoint action of $$G_n$$ preserves $$\mathrm{Lie}(G)$$, by passing to the limit we get that $$\mathrm{Lie}(H)$$ is an ideal of $$\mathrm{Lie}(G)$$.
Now a bit of (elementary ?) Lie theory should give you that $$\mathrm{Lie}(G_n) = \mathrm{Lie}(H)$$ for $$n$$ large enough. Let $$H$$ be the connected subgroup with Lie algebra $$\mathrm{Lie}(H)$$. One concludes that $$G_n/H$$ is a sequence of discrete groups approximating $$G/H$$. By your second reference, $$G/H$$ is abelian, which proves your conjecture.