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?