A height gap in $GL_d(\overline{\mathbb{Q}})$ and almost laws

TL;DR

This paper provides a new, shorter proof for a lower bound on the normalized height of generating sets in $GL_d(ar{Q})$, using properties of almost laws in compact Lie groups, linking arithmetic size with group structure.

Contribution

The paper introduces a novel, concise proof of Breuillard's height gap theorem, leveraging the concept of almost laws in compact Lie groups.

Findings

Established a lower bound for normalized height in $GL_d(ar{Q})$

Connected almost laws in Lie groups to arithmetic properties of matrix groups

Provided a simplified proof technique for a key result in arithmetic group theory

Abstract

E. Breuillard showed that finite subsets $F$ of matrices in $GL_d(\overline{\mathbb{Q}})$ generating non-virtually solvable groups have normalized height $\widehat{h}(F) \ge \epsilon_d$, for some positive $\epsilon_d >0$. The normalized height $\widehat{h}(F)$ is a measure of the arithmetic size of $F$ and this result can be thought of as a non-abelian analog of Lehmer's Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.