Bottom of the Length Spectrum of Arithmetic Orbifolds

Abstract

A. We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from 1. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold Γ\ by showing the existence of a positive constant ( ) > 0 such that squares of lengths of closed geodesics shorter than must be pairwise linearly dependent over Q.