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

Abstract

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 $G{L}_n(\bar{\mathbb{Q}})$ 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 $S{L}_n(\bar{\mathbb{Q}}),$ where $\epsilon =\epsilon (n)>0$ is an absolute constant.

References in corpus (1)

• Infinite groups with large balls of torsion elements and small entropy

Cited by in corpus (1)

• A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups