Temperedness of locally symmetric spaces: The product case

Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $\mathrm{\Gamma }$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $\mathrm{\Gamma }\mathrm{\backslash }X$ is related to the asymptotic growth of $\mathrm{\Gamma }$ in the two direction defined by the two factors. We obtain that $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$ is tempered for large class of $\mathrm{\Gamma }$.