Absence of principal eigenvalues for higher rank locally symmetric spaces

T. Weich, L.L. Wolf, ArXiv:2205.03167 (2022).

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Abstract
Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $\Gamma\backslash G/K$ of higher rank. We derive dynamical assumptions on the $\Gamma$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.
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arXiv:2205.03167
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Weich T, Wolf LL. Absence of principal eigenvalues for higher rank locally symmetric  spaces. arXiv:220503167. Published online 2022.
Weich, T., & Wolf, L. L. (2022). Absence of principal eigenvalues for higher rank locally symmetric  spaces. In arXiv:2205.03167.
@article{Weich_Wolf_2022, title={Absence of principal eigenvalues for higher rank locally symmetric  spaces}, journal={arXiv:2205.03167}, author={Weich, Tobias and Wolf, Lasse Lennart}, year={2022} }
Weich, Tobias, and Lasse Lennart Wolf. “Absence of Principal Eigenvalues for Higher Rank Locally Symmetric  Spaces.” ArXiv:2205.03167, 2022.
T. Weich and L. L. Wolf, “Absence of principal eigenvalues for higher rank locally symmetric  spaces,” arXiv:2205.03167. 2022.
Weich, Tobias, and Lasse Lennart Wolf. “Absence of Principal Eigenvalues for Higher Rank Locally Symmetric  Spaces.” ArXiv:2205.03167, 2022.

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