Sarnak’s spectral gap question

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...

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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 151; no. 1; pp. 171 - 179
Main Authors Kelmer, Dubi, Kontorovich, Alex, Lutsko, Christopher
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.12.2023
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Analysis
Dynamical Systems and Ergodic Theory
Eigenvectors
Functional Analysis
Mathematics
Mathematics and Statistics
Partial Differential Equations
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Summary: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.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-023-0322-z