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

Abstract

E. Breuillard showed that finite subsets $F$ of matrices in ${\mathrm{GL}}_d(\overline{\mathbb{Q}})$ generating non-virtually solvable groups have normalized height $\hat{h}(F)\ge {\epsilon }_d$, for some positive ${\epsilon }_d>0$. The normalized height $\hat{h}(F)$ 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.