A height gap in GLd(Q̄) and almost laws

E. Breuillard showed that finite subsets F{-.75pt} of matrices in GL_{d{-.75pt}}({-.75pt}Q̄{-.75pt}) generating non-virtually solvable groups have normalized height ĥ(F)≥ε_d , for some positive εd>0 . The normalized height ĥ(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.