Lax-Phillips orbit counting in higher rank

Abstract

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

14 pages, comments welcome; Accepted for publication in BLMS