A Height Gap Theorem For Finite Subsets Of GL_d(\bar{Q}) and Non Amenable Subgroups

We 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 $\hat{h}(F)>ε$ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $ε=ε(n)>0$ is an absolute constant.

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