Temperedness of locally symmetric spaces: the product case

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