Temperedness of locally symmetric spaces: the product case

Let X=X₁× X₂ X = X 1 × X 2 be a product of two rank one symmetric spaces of non-compact type and Γ Γ a torsion-free discrete subgroup in G₁× G₂ G 1 × G 2 . We show that the spectrum of Γ (X₁× X₂) Γ \ ( X 1 × X 2 ) is related to the asymptotic growth of Γ Γ in the two directions defined by the two factors. We obtain that L²(Γ (G₁ × G₂)) L 2 ( Γ \ ( G 1 × G 2 ) ) is tempered for a large class of Γ Γ .