# Transitive group actions and homogeneous spaces

Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is homeomorphic to $G/G_x$, if $G$ is $\sigma$ compact. (Proposition 2.44, page 55)

Are there counter examples? Theo Buehler's answer in the comments: any compact infinite group $X=K$ and $K_d$ being the $K$ with the discrete topology still acts continuously and transitively, but $X \neq K_d$.

I have a certain issue with a proof, which assumes something similar without mentioning $\sigma$ compactness at all. The statement for a closed subgroup $H$ and a compact subgroup $K$ in a locally compact Hausdorff group $G$ with $G = KH$, we have an isomorphism $G \cong K \times H / K \cap H$. They use that $K \times H$ acts transitively on $G$ with $K \cap H$ as isotropy group, but do not mention any $\sigma$ compactness condition. Theo Buehler's counterexample does not apply here, since the groups carry the subspace topology. But why is this argument correct?

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Re: first question. Let $G$ be locally compact non-discrete and let $G_d$ be $G$ equipped with the discrete topology. Let $G_d$ act on $G$ via left translations. –  t.b. Aug 10 '11 at 10:33
For the 2nd question, if $G$ really only locally compact, or locally compact Hausdorff? –  John M Aug 10 '11 at 11:30
The group is locally compact Hausdorff. Sorry for that imprecision, but the reference assumes per definition that locally compact groups are Hausdorff, but why are you asking? –  plusepsilon.de Aug 10 '11 at 13:00
Hi I was just asking to be sure. For the 2nd result re: G=KH etc, it does feel like we're missing some additional hypothesis here. Maybe Theo can weigh in, but I think that rather than sigma-compact, I think you can get away with 2nd countable - perhaps that's in your 2nd result somewhere? Do you have a reference for it? –  John M Aug 10 '11 at 14:16
ah...in the proof they've "identified", i.e. as sets, not necessarily as isomorphic or homeomorphic. –  John M Aug 10 '11 at 14:43

## 1 Answer

You didn't specify it, but I think the only way to interpret the second question is the following:

The group $H \times K$ acts on $G$ by $(h,k)g = hgk^{-1}$. Since $G = HK = HK^{-1}$ this action is transitive and the stabilizer of the neutral element is $L = (l,l)$ with $l \in H \cap K$ and clearly $L \cong H \cap K$ is compact as $H$ is closed and $K$ is compact in $G$. Then $G$ is indeed homeomorphic to $(H \times K) \,/\,L$ because we have:

Claim. The action of $H \times K$ on $G$ is proper.

For background and references on proper actions, see my answer on MO. Given this, we have that the orbit map $H \times K \to G$ given by $(h,k) \mapsto hk^{-1} = (h,k)e$ is proper and therefore the map $(H \times K) \,/\,L \to G$ is a proper bijection, hence a homeomorphism since proper maps are closed.

To prove that the action is indeed proper, let $C_1, C_2 \subset G$ be compact. We need to show that $\{(h,k) \in H \times K\,:\,((h,k)C_1) \cap C_2 \neq \emptyset\}$ is relatively compact in $H \times K$. However, this is clear because it is contained in the set $C \times K$ where $C = \overline{\{h \in H\,:\,(hC_1K) \cap C_2 \neq \emptyset\}} \subset H$ is obviously compact.

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This discussion is reminiscent of another either here or MO... Note, for $G$ transitive on $X$, $G_x$ the isotropy group of a point $x\in X$, if we already assume that $G=HG_x$ and that $X\approx H/(H\cap G_x)$, we've assumed away one serious issue, which would need Baire category, I think. Beyond that, countably-based, locally-compact, Hausdorff seems needed for $G$, and $G\times X\rightarrow X$ continuous, and no assumption that the isotropy group is either compact or co-compact. –  paul garrett Aug 10 '11 at 20:28
Oh, yes! The 'apocryphal lemma!' Now that you mention it, of course... I should have remembered :) I think it is this discussion you have in mind. Your solenoid notes are still on my big I should read that more attentively stack. –  t.b. Aug 10 '11 at 20:38
Ha! @Theo B. Hahaha! Indeed. And, kind of you to think of solenoid notes... My disappointment is that "kids these days" read "ur-solenoid" (the adelic sequel to the 2-adic version) as "your"... and I get email and paper-mail spam about winding-coil products and seminars. :) –  paul garrett Aug 10 '11 at 23:15
... and why does no one remember Mesopotamia, Ninevah, not to mention Sumer... anymore? –  paul garrett Aug 10 '11 at 23:22
... What can I say? Things get all the more mysterious when observing that people might know what is meant by writings on the wall, mene mene tekel, u-parsin and relate Nebukhadnezzar to either Babylon or a space-ship-like device in a successful movie named after a linear algebra gadget... Not to mention that many mathematicians know what an Eilenberg-Mac Lane space is but don't know what rôle "your" Solenoid played in the history of category theory... As a side note: googling "ur-solenoid" gives me your notes as a first hit, and the second hit says "ur solenoid is buzzing"... –  t.b. Aug 11 '11 at 1:25