Linear groups and heights

By the Tits alternative, every finitely generated linear group is either virtually solvable or contains a free group of rank 2. More recently, Breuillard established a stronger version, the so-called uniform Tits alternative for linear groups (where uniformity refers to independence over the base field), using methods from diophantine geometry. As a consequence, he obtained a uniform growth gap theorem for the exponential growth rate of non virtually solvable linear groups. This should be viewed in the context of the growth conjecture on linear groups, which is related to the Lehmer conjecture in algebraic number theory. In this lecture, we will discuss the work by Breuillard on the relation between growth of groups and the Lehmer conjecture. In particular, we will recall the necessary background from geometric group theory and diophantine geometry.