# Metrically homogeneous spaces as inverse limits

Let $$(X,d)$$ be a locally compact, separable, connected and $$\sigma$$-compact metric space such that the group of isometries $$G$$ acts transitively on $$X$$. The question is the following:

Is $$X$$ topologically the inverse limit of homogeneous manifolds?

We call a manifold homogeneous if it is of the form $$L/C$$, where $$L$$ is Lie and $$C$$ is a closed subgroup.

This seems to be a consequence of the solution to Hilbert's fifth problem, but I am skeptic since I have not been able to find such a result in the literature. E.g. in arxiv.org/1301.5114 they derive related results in the case where $$X$$ is locally contractible. So, I a have the suspicion that I might be missing something. The tentative proof goes as follows:

We endow $$G$$ with a suitable topology making it a locally compact topological group. Since $$X$$ is $$\sigma$$-compact, if we have $$X=\cup C_n$$ we may topologize it using the following family of pseudometrics: $$d_n(f,g)=\max_{x\in C_n}\{\, d(f(x),g(x)),\,d(f^{-1}(x),g^{-1}(x))\,\}$$ Let $$x\in X$$ and let $$S$$ be its stabilizer (which is compact). Then $$X\cong G/S$$. Since $$G$$ is a locally compact group, we have that there is an open subgroup $$G_o$$ and a sequence of decreasing compact normal subgroups $$K_n$$, contained in arbitrarily small neighbourhoods of $$\text{id}_X$$ and such that $$K_n\backslash G_0$$ is Lie. Since $$X$$ is connected, we have that $$G_0(X)=X$$, so w.l.o.g. we may assume $$G_0=G$$.

After this discussion, what remains is to show that $$G/S$$ is the inverse limit of the spaces $$K_n\backslash G/S$$, which seems to be straightforward. Since the action of $$S$$ on $$K_n\backslash G$$ is by right multiplication and $$S$$ is compact, the action factors through the quotient $$(K_n\cap S)\backslash S$$, which can be identified with a compact subgroup of $$K_n\backslash G$$. Thus the space $$K_n\backslash G/S$$ is a quotient of a Lie group by a closed subgroup, i.e. a homogeneous manifold.

Assuming that the result is correct, can the assumptions of connectedness and $$\sigma$$-compactness be dropped?

• You want $X$ to be a projective limit of homogeneous manifolds, but just a projective limit in the topological category (ignoring any metric or any compatibility between the manifolds being homogeneous). Probably the proofs should provide stronger statements. – YCor Feb 11 at 13:44