Temperedness of locally symmetric spaces: The product case - Weich, Tobias; Wolf, Lasse Lennart (Geom Dedicata)
Let $X=X_1× X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1× G_2$. We show that the spectrum of $Γ \backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ \backslash G)$ is tempered for large class of $Γ$.

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces - Lutsko, Christopher; Weich, Tobias; Wolf, Lasse Lennart (arXiv:2402.02530)
Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $Γ < G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(Γ\backslash G)$ in this general setting.

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