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Sam Edwards - The Bottom of the $L^2$ Spectrum of Higher-rank Locally Symmetric Spaces
· 21.06.2023 · 17:50:34 ··· MiTTwoch ⭐ 0 🎬 0
📺 Institut des Hautes Etudes Scientifiques (IHES)
For a rank one geometrically finite locally symmetric space Γ\X, the bottom of the $L^2$ spectrum of the Laplace operator is a simple eigenvalue corresponding to a positive eigenfunction if and only if the critical exponent of Γ is strictly greater than half the volume entropy of X. In particular, there exist infinite volume rank one locally symmetric spaces with square integrable positive Laplace eigenfunctions. In contrast, a higher-rank symmetric space Γ\X without rank one factors has a square integrable positive Laplace eigenfunction if and only Γ is a lattice. We will explain some aspects of the connection between square integrability of positive Laplace eigenfunctions and Patterson-Sullivan and Bowen-Margulis-Sullivan measures in the higher-rank setting. Based on joint work with Oh and Fraczyk-Lee-Oh.
Sam Edwards (Durham University)
· 21.06.2023 · 17:50:34 ··· MiTTwoch
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