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First of all, let's agree on a definition. Let $X$ be a topological space and let $\Gamma$ be a topological group.

A continous group action $\alpha:\Gamma \times X\to X$ is properly discontinous iff for every compact $K$ we have $K\cap \gamma K\neq \varnothing$ only for a finite number of $\gamma \in \Gamma$.

Let $X$ be a $T_2$, second countable, locally compact space and let $G$ be a subgroup of the homeomorphism group of $X$ (equipped with the compact-open topology). Let's suppose that $G$ acts on $X$ transitively and with compact stabilizers: $$\alpha:G\times X\to X.$$ Clearly this induces an action of every subgroup $\Gamma \leq G$ on $X$, simply given by $\alpha|_{\Gamma \times X}$. Now here comes the request of the problem:

Prove that if $\Gamma \leq G$ is a discrete subgroup of the group of the homeomorphism of $X$, then $\alpha|_{\Gamma \times X}$ is a properly discontinous action.

I really don't know where to start with this problem. I feel just that there is so much going on here. I tried even to treat the case $\Gamma=G$ first to simplify things but I really don't know how to approach this.

Kandinskij
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  • What makes you think that this claim is true? – Moishe Kohan Apr 18 '23 at 20:14
  • The fact that it was in my sheet of exercises... (it's not intended to sound rude, I'm just being honest). – Kandinskij Apr 18 '23 at 20:15
  • Note "properly discontinuous" can be defined differently – FShrike Apr 18 '23 at 20:29
  • What does "compact stabilisers" mean? – FShrike Apr 18 '23 at 20:34
  • That the stabilizers of the group actions are compact subspaces of $G$ – Kandinskij Apr 18 '23 at 20:36
  • I thought I removed this comment: It was written when I did not read the question carefully. – Moishe Kohan Apr 18 '23 at 22:05
  • I think a place to start looking is at the fact that $\Gamma$ is discrete under the compact-open topology. I.e., for any compact $K$ there is a neighborhood $V$ of $\gamma K$ for which $\gamma$ is the only element of $\Gamma$ which carries $K$ into $V$. – Paul Sinclair Apr 19 '23 at 14:42
  • I forgot in the prevous comment that for given $K, V$, ${\gamma \in \Gamma : \gamma K\subseteq V}$ is only a sub-basis element of the compact-open topology. So the best you can say for any given $\gamma$ is that there is a finite collection of compact sets $K_i$ and neighborhoods $V_i$ of $\gamma K_i$ for which $\gamma$ is the only element of $\Gamma$ that carries every $K_i$ into $V_i$. Still, I think this is the key to your solution. – Paul Sinclair Apr 19 '23 at 15:20
  • Do you already know that each orbit map induces a homeomorphism $G/G_x\to X$? – Moishe Kohan Apr 21 '23 at 16:07
  • @MoisheKohan I know that this is a bijection by orbit-stabilizer theorem, but I don't know how to prove that it's an homeomorphism. – Kandinskij Apr 21 '23 at 16:09
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    Then take a look at my proof here. – Moishe Kohan Apr 21 '23 at 16:13
  • @MoisheKohan Got it! – Kandinskij Apr 21 '23 at 17:04

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