Arithmetic Groups and the Lehmer Conjecture

TL;DR

This paper establishes a connection between the uniform discreteness of cocompact lattices in higher rank semisimple Lie groups and a weak form of Lehmer's conjecture, providing new insights into their equivalence.

Contribution

It proves the equivalence between uniform discreteness of certain lattices and a weak Lehmer's conjecture, extending previous results and including a survey of related topics.

Findings

Uniform discreteness of lattices is equivalent to a weak Lehmer's conjecture.

Provides a survey of related results and conjectures.

Extends known results in higher rank semisimple Lie groups.

Abstract

We generalize a result of Sury and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.