Absence of principal eigenvalues for higher rank locally symmetric spaces

TL;DR

This paper generalizes classical results on the absence of principal eigenvalues for Laplacians on hyperbolic surfaces to higher rank locally symmetric spaces, using dynamical assumptions related to group actions.

Contribution

It extends the non-existence of principal eigenvalues to a broader class of higher rank symmetric spaces under specific dynamical conditions.

Findings

Absence of principal eigenvalues for higher rank locally symmetric spaces.

Dynamical conditions on group actions imply spectral gaps.

Applicable to non-compact quotients by Anosov subgroups.

Abstract

Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $\Gamma\backslash G/K$ of higher rank. We derive dynamical assumptions on the $\Gamma$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.