$L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces

Lasse L. Wolf; Hong-Wei Zhang

doi:10.1090/proc/16935

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arXiv · 2311.11770

$L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces

Lasse L. Wolf·Hong-Wei Zhang

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Abstract

In this short note we observe, on locally symmetric spaces of higher rank, a connection between the growth indicator function introduced by Quint and the modified critical exponent of the Poincar\'e series equipped with the polyhedral distance. As a consequence, we provide a different characterization of the bottom of the $L^2$-spectrum of the Laplace-Beltrami operator in terms of the growth indicator function. Moreover, we explore the relationship between these three objects and the temperedness.

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BibTeX RIS

Lasse L. Wolf, Hong-Wei Zhang. 2023-11-20. $L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces. https://doi.org/10.1090/proc%2F16935

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