Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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

1 Answer 1

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 .

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.