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International audienceWe show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\bar{\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\hat{h}(F)>\epsilon $ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $\epsilon =\epsilon (n)>0$ is an absolute constant

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