Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces

Jean-Philippe Anker (1) , Hong-Wei Zhang (1, 2)

1 IDP - Institut Denis Poisson
2 UGENT - Universiteit Gent = Ghent University

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.

Mots clés: Poincaré series; Critical exponent; Heat kernel; Locally symmetric space; $L^2$ spectrum

Domaines: Mathématiques [math]; Théorie spectrale [math.SP]

Jean-Philippe Anker, Hong-Wei Zhang. Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces. Geometriae Dedicata, 2022, 216 (1), paper 3. ⟨hal-02865274v2⟩