Salem numbers and arithmetic hyperbolic groups

TL;DR

This paper establishes a link between Salem numbers and the geometry of arithmetic hyperbolic groups, providing bounds on geodesic lengths and connecting conjectures in number theory and hyperbolic geometry.

Contribution

It reveals a direct relationship between Salem numbers and hyperbolic group elements, and proves the equivalence of a geometric conjecture with Lehmer's conjecture.

Findings

A sharp lower bound for closed geodesic lengths in noncompact arithmetic hyperbolic orbifolds.

Proved the equivalence of the 'short geodesic conjecture' with Lehmer's conjecture.

Established a connection between Salem numbers and translation lengths in hyperbolic groups.

Abstract

In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for Salem numbers.