Spectral bounds, orbit growth, and temperedness of locally symmetric spaces.

Consider a semi-simple Lie group, $G$ and a discrete torsion-free subgroup $\Gamma < G$. When $G$ has rank one, there is an important, well-understood connection between the growth rate of elements of $\Gamma$-orbits, bounds on the spectrum, and temperedness, or more generally properties of the unitary $L^2$ representation (due, in part, to work of Elstrodt and Patterson). In this talk I will present an extension of this connection to higher rank.  This is joint work with Tobias Weich and Lasse Wolf.