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Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces
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abstract
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 $\mathfrak{e}_{6(-26)}$.
Forward citations
Cited by 2 Pith papers
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Zariski density of discrete subgroups via critical exponents and unitary representations
Discrete subgroups with critical exponent sufficiently close to the volume growth entropy of the symmetric space are Zariski dense, generalizing Borel's density theorem.
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The limit cone and bounds on the growth indicator function
If the limit cone of a discrete subgroup avoids two unrelated Weyl-chamber facets, Quint's growth indicator is at most the half-sum of positive roots, so L²(Γ\G) is tempered.
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