Bottom of the Length Spectrum of Arithmetic Orbifolds

TL;DR

This paper characterizes the discreteness of arithmetic lattices in Lie groups via Salem numbers and explores the structure of the shortest closed geodesics in arithmetic orbifolds, revealing linear dependencies among their lengths.

Contribution

It establishes a link between Salem numbers and uniform discreteness of arithmetic lattices and analyzes the structure of the length spectrum of arithmetic orbifolds.

Findings

Cocompact arithmetic lattices are uniformly discrete iff Salem numbers are bounded away from 1.

Existence of a positive constant ensuring short geodesic lengths are linearly dependent over Q.

Insights into the structure of the 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 $\mathbb Q$.