A height gap theorem for finite subsets of $\mathrm{GL}_{d}(\overline{\Bbb{Q}})$ and nonamenable subgroups

Abstract

We introduce a conjugation invariant normalized height $\widehat{h}(F)$ on finite subsets of matrices $F$ in $\mathrm{GL}_{d}(\overline{\Bbb{Q}})$ and describe its properties. In particular, we prove an analogue of the Lehmer problem for this height by showing that $\widehat{h}(F)>\varepsilon $ whenever $F$ generates a nonvirtually solvable subgroup of $\mathrm{GL}_{d}(\overline{\Bbb{Q}}),$ where $\varepsilon =\varepsilon (d)>0$ 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.