A height gap theorem for finite subsets of GL_d(Q) and nonamenable subgroups

Abstract

We introduce a conjugation invariant normalized heightb(F ) on nite subsets of matrices F in GLd(Q) and describe its properties. In particular, we prove an analogue of the Lehmer problem for this height by showing that b(F ) > whenever F generates a nonvirtually solvable subgroup of GLd(Q); where = (d) > 0 is an absolute constant. This can be seen as a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. As an application we prove a uniform version of the classical Burnside-Schur theorem on torsion linear groups. In a companion paper we will apply these results to prove a strong uniform version of the Tits alternative.