A group-invariant version of Lehmer's conjecture on heights

TL;DR

This paper proves an equivariant version of Lehmer's conjecture on heights, extending previous special cases and classifying all groups with finite, non-empty orbits where non-zero elements lie on the unit circle.

Contribution

It introduces a generalized, group-invariant form of Lehmer's conjecture and classifies all groups satisfying specific orbit finiteness conditions.

Findings

Proved an equivariant version of Lehmer's conjecture.

Extended previous special cases to a complete classification.

Identified all groups with finite, non-empty orbits on the unit circle.

Abstract

We state and prove an equivariant version of Lehmer's conjecture on heights, generalizing papers by Zagier (1993) and Dresden (1998) which are special cases of this theorem. We also extend their three cases to a full classification of all groups satisfying the condition that the set of all orbits for which every non-zero element lies on the unit circle is finite and non-empty.