Temperedness of locally symmetric spaces: the product case

Abstract

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