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Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.

Consider the following conditions on $G$:

(1) $G$ acts properly on $X$, i.e. any two points $x$ and $y$ in $X$ have neighborhoods $U_x$ and $U_y$ such that there are only a finite number of group elements $g \in G$ with $g(U_x)$ meeting $U_y$.

(2) $G$ is a discrete subgroup of $\mathrm{ISO}(X)$.

(3) The orbit $Gx$ is a discrete subset of $X$ for all $x \in X$.

My question: Is (1) equivalent with (2) or is (2) equivalent with (3), or neither? Does anything change if one assumes also that $X$ is CAT(0) and/or $G$ acts cocompactly.

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Could you give some context for the question? Did it arise in your research, or are you just curious? As Misha says below, it looks a lot like a homework question. I'm going to vote to close, although the question could be suitable for MO with more context. – HJRW Mar 7 '12 at 19:55
It was not a homework question, just a question I asked myself. But I see now that the answer is not very difficult. I was just forgetting about Arzela-Ascoli theorem. So thanks for the hint Misha. – unknown Mar 8 '12 at 8:22
up vote 0 down vote accepted

This is a nice homework problem for a graduate class in metric geometry or geometric group theory right after the students learned the definition of uniform convergence on compacts and the Arzela-Ascoli theorem. (No, the students do not need to know what CAT(0) spaces are.)

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@Misha: this should be a comment (since you are not actually answering the question). – Igor Rivin Mar 7 '12 at 14:26
Igor, I feel that this is a hwk question, so it would be best to give a hint rather than an answer. Being new to MO, I am not quite sure what is the policy on this... – Misha Mar 7 '12 at 16:09
Misha - the usual policy is to write a polite comment to that effect, possibly refer the asker to math.stackexchange (with the caveat that if it's a homework question then that should be clearly stated) and vote to close (if you have 3000+ reputation, as you no doubt will before long). – HJRW Mar 7 '12 at 19:51

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