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Bottom of the \(L^2\) spectrum of the Laplacian on locally symmetric spaces

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Abstract

We estimate the bottom of the \(L^2\) spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank.

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Notes

  1. The symbol \(f\asymp g\) between two non-negative expressions means that there exist constants \(0<A\le B<+\infty \) such that \(Ag\le f\le Bg\).

  2. As observed by the referee, Lemma 3 still holds without the torsion-free assumption, provided that \(\gamma \) runs through \(\varGamma \backslash (\varGamma \cap {K})\).

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Acknowledgements

The authors are grateful to the referee for checking carefully the manuscript and making several helpful suggestions of improvement. The second author acknowledges financial support from the University of Orléans during his Ph.D. and from the Methusalem Programme Analysis and Partial Differential Equations during his postdoc stay at Ghent University.

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Authors and Affiliations

  1. Institut Denis Poisson (UMR 7013), Université d’Orléans, Université de Tours & CNRS, Bâtiment de Mathématiques, Rue de Chartres, B.P. 6759, 45067, Orléans cedex 2, France

    Jean-Philippe Anker & Hong-Wei Zhang

  2. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, Ghent, 9000, Belgium

    Hong-Wei Zhang

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  1. Jean-Philippe Anker
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  2. Hong-Wei Zhang
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Correspondence to Jean-Philippe Anker.

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In memory of Michel Marias (1953–2020), who introduced us to locally symmetric spaces.

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Anker, JP., Zhang, HW. Bottom of the \(L^2\) spectrum of the Laplacian on locally symmetric spaces. Geom Dedicata 216, 3 (2022). https://doi.org/10.1007/s10711-021-00662-7

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  • DOI: https://doi.org/10.1007/s10711-021-00662-7

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