  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.
