Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces

Abstract: Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $\Gamma < G$ we prove a precise connection between growth rates of the group $\Gamma$, 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 $L^2(\Gamma\backslash G)$ in this general setting.