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

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