Geodesics, volumes and Lehmer's conjecture

TL;DR

This paper explores the connections between the geometric properties of hyperbolic manifolds and Lehmer's conjecture on Mahler measures, aiming to shed light on number theory and geometric topology.

Contribution

It investigates the relationship between systoles, volumes, and Lehmer's conjecture, providing new insights into their interplay in hyperbolic geometry.

Findings

Identifies links between systoles and Mahler measures.

Provides evidence supporting Lehmer's conjecture in geometric contexts.

Suggests implications for hyperbolic volume bounds.

Abstract

In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.