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 Salem numbers are uniformly bounded away from 1 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 ∖ X \Gamma \backslash X by showing the existence of a positive constant δ ( X ) > 0 \delta (X)>0 such that squares of lengths of closed geodesics shorter than δ \delta must be pairwise linearly dependent over Q \mathbb {Q} .