Temperedness of locally symmetric spaces: the product case

Abstract

AbstractLet $$X=X_1\times X_2$$ X = X 1 × X 2 be a product of two rank one symmetric spaces of non-compact type and $$\Gamma $$ Γ a torsion-free discrete subgroup in $$G_1\times G_2$$ G 1 × G 2 . We show that the spectrum of $$\Gamma \backslash (X_1\times X_2)$$ Γ \ ( X 1 × X 2 ) is related to the asymptotic growth of $$\Gamma $$ Γ in the two directions defined by the two factors. We obtain that $$L^2(\Gamma \backslash (G_1 \times G_2))$$ L 2 ( Γ \ ( G 1 × G 2 ) ) is tempered for a large class of $$\Gamma $$ Γ .