Abstract
We answer in the affirmative a question of Sarnak’s from 2007, confirming that the Patterson–Sullivan base eigenfunction is the unique square-integrable eigenfunction of the hyperbolic Laplacian invariant under the group of symmetries of the Apollonian packing. Thus the latter has a maximal spectral gap. We prove further restrictions on the spectrum of the Laplacian on a wide class of manifolds coming from Kleinian sphere packings.
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Acknowledgement
We thank Peter Sarnak for bringing this problem to our attention, and comments on an earlier draft.
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Dedicated to Peter Sarnak with admiration and trepidation
Kelmer is partially supported by NSF CAREER grant DMS-1651563.
Kontorovich is partially supported by NSF grant DMS-1802119 and BSF grant 2020119.
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Kelmer, D., Kontorovich, A. & Lutsko, C. Sarnak’s spectral gap question. JAMA 151, 171–179 (2023). https://doi.org/10.1007/s11854-023-0322-z
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DOI: https://doi.org/10.1007/s11854-023-0322-z