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. 



Discrete subgroups of isometry group of proper metric space
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.) 


