Lax-Phillips orbit counting in higher rank

Abstract

Given a discrete lattice, $\Gamma < \operatorname{SL}_m(\mathbb{R})$, and a base point $o \in \mathbb{R}^m$, let $N_\Gamma(T)$ denote the number of points in the orbit $o \cdot \Gamma $ whose (Euclidean) length is bounded by a growing parameter, $T$. We demonstrate an abstract spectral method \`a la Lax-Phillips, capable of obtaining strong asymptotic estimates for $N_\Gamma(T)$, and compare and contrast it with other methods.