This research proves the absence of new eigenvalues in finite coverings of Riemannian manifolds, suggesting implications for fundamental groups.
We prove the non-existence of new eigenvalues in $[0,Λ]$ for specific and random finite coverings of a complete and connected Riemannian manifold M with Ricci curvature bounded from below, where $Λ$ is any positive number below the essential spectrum of M and the spectrum of the universal cover of M, provided the representation theory of the fundamental group of M satisfies certain conditions.
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Ballmann et al. (2025) studied this question.
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