Geodesic
A height gap theorem for finite subsets of $\mathrm{GL}_{d}(\overline{\Bbb{Q}})$ and nonamenable subgroups
Emmanuel Breuillard1
1 Laboratoire de Mathématiques<br/>Université Paris Sud<br/> 91405 Orsay cedex<br/> France
Annals of mathematics, Tome 174 (2011) no. 2, pp. 1057-1110.

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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.
DOI : 10.4007/annals.2011.174.2.7

Emmanuel Breuillard 1

1 Laboratoire de Mathématiques<br/>Université Paris Sud<br/> 91405 Orsay cedex<br/> France 
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Emmanuel Breuillard. A height gap theorem for finite subsets of $\mathrm{GL}_{d}(\overline{\Bbb{Q}})$ and nonamenable subgroups. Annals of mathematics, Tome 174 (2011) no. 2, pp. 1057-1110. doi : 10.4007/annals.2011.174.2.7. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.7/

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