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

Given a real semisimple connected Lie group $G$ and a discrete subgroup $\mathrm{\Gamma }<G$ we prove a precise connection between growth rates of the group $\mathrm{\Gamma }$, 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 $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$ in terms of Quint’s growth indicator function. As an application of our sharp polyhedral bounds we prove temperedness of $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$ for all Borel Anosov subgroups $\mathrm{\Gamma }$ in higher rank Lie groups $G$ not locally isomorphic to ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{K}),\mathbb{K}=\mathbb{R},\text{C},\mathbb{H},$ or ${\mathfrak{e}}_{6(-26)}$.