Bottom of the Length Spectrum of Arithmetic Orbifolds

Abstract

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 $\Gamma\backslash X$ by showing the existence of a positive constant $\delta(X)>0$ such that squares of lengths of closed geodesics shorter than $\delta$ must be pairwise linearly dependent over $\mathbb Q$.