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

TL;DR

This paper provides a new characterization of the bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces, extending known results from rank one to higher rank and improving previous bounds.

Contribution

It introduces a higher rank analog of a spectral characterization, enhancing understanding of the Laplacian's spectrum on locally symmetric spaces.

Findings

Estimates the bottom of the $L^2$ spectrum using Poincaré series exponents

Generalizes rank one spectral characterizations to higher rank

Improves bounds over previous higher rank results

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 Poincar\'e 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.