Suppose $X\_1$ and $X\_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G\_i=Isom(X\_i)$. Then $G\_1\times G\_2\subset Isom(X\_1\times X\_2)$.


Suppose $X\_1\times X\_2$ is not compact and let $\Gamma\subset G\_1\times G\_2$ act cocompactly on $X\_1\times X\_2$. In general the projections of $\Gamma$ to $G\_1$ and $G\_2$ are not discrete.


Example: Let $X\_1=S^1$, $X\_2=\mathbb R$, and $f=(\alpha,\tau)$ where $\alpha$ is an irrational rotation an $\tau(x)=x+1$. The group generated by $f$ is discrete but its projection to $G\_1$ is not discrete. 


Is there a characterizatrion (or a sufficient condition) of the situation where both projections of $\Gamma$ to $G\_1$ and $G\_2$ are discrete?



discrete subgroups of the isometries of a product
This is presumably a hard question, see e.g. <http://download.springer.com/static/pdf/747/art%253A10.1007%252FBF02698916.pdf?auth66=1382759591_9b0973395ce39560e6d6963480bf665c&ext=.pdf>


