Local-global Phenomenon in Thin Matrix Group Orbits

Abstract

We describe a few natural arithmetic problems arising in completely unrelated settings, such as the curvature problem on the Apollonian gaskets and their kind, Zaremba’s problem on the Monte Carlo method, and the length spectrum of sonic hyperbolic surfaces, which turn out to have a common formulation involving “thin” matrix group orbits. Recent progress comes from a synthesis of modern techniques including harmonic analysis, algebra, geometry and dynamics. We describe the problems, the progress and some of the techniques invoIved.