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So I'm aware that the orbit-stabilizer theorem does not hold for arbitrary spaces with a transitive action by a topological group, but I wonder if it works in the following situation.

Let $G$ be a totally disconnected, locally compact Hausdorff topological group and $X$ a topological space satisfying the same conditions (I would call such things $\ell$-groups and $\ell$-spaces respectively). If $G$ acts transitively on $X$ and $x \in X$ is any point there is an obvious $G$-equivariant continuous bijection $G/G_x \to X$, where $G_x$ denotes the stabilizer of $x$ in $G$. Can we conclude, in this situation, that $G/G_x \to X$ is a homeomorphism? If not, what further conditions do we need to impose? Notice that this is true if $G/G_x$ is compact, since a continuous bijection of compact Hausdorff spaces is a homeomorphism.

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Well, I don't have much intuition about totally disconnected groups and spaces, but one general criterion that may (or may not) be useful is if the action is proper, i.e. the map $G \times X \to X \times X$, $(g,x) \mapsto (gx,x)$ is proper in the sense that pre-images of compact sets are compact then the map $G/G_x \to Gx$ is a homeomorphism, see this MO-thread for some basics on proper actions and references. – t.b. Jun 23 '11 at 20:02
up vote 5 down vote accepted

This is true for G a locally compact, Hausdorff topological group, and X locally compact, Hausdorff, with a countable local basis. This "apocryphal lemma" appears many places, but is easily misplaced.

I reproduced the usual argument in an appendix in the "Solenoids" class notes on my modular forms course page, here .

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Thanks, I was hoping I could get away without technical hypotheses like second countability, but this is nice enough. – Justin Campbell Jun 23 '11 at 21:07
+1, it's exciting to see Prof. Garrett active on this site. (BTW, now that the lemma has been written down in a canonical place, is it still "apocryphal"?) – Pete L. Clark Jun 23 '11 at 21:23
Thanks, Pete! (I've always called it "the apocryphal lemma", but could change, since that was never a very useful label. "Orbit stabilizer theorem"?) – paul garrett Jun 23 '11 at 21:56
Assuming $G$ is locally compact second countable and $X$ is Polish, a refinement of this is proved as Theorem 2.1.14 in Ergodic Theory and Semisimple Groups by Robert J. Zimmer. The references given there are Glimm (locally compact case) and Effros. @Pete, maybe you're interested. – t.b. Jun 23 '11 at 22:36

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