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

Abstract We introduce a conjugation invariant normalized height hˆ(F) on finite 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 hˆ(F)>ε whenever F generates a nonvirtually solvable subgroup of GLd(ℚ⎯⎯⎯), where ε=ε(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.

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### 数学年刊（Annals of Mathematics）

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