A height gap theorem for finite subsets of GL(d)((Q)over-bar) and nonamenable subgroups

作者：Breuillard Emmanuel^*

来源：Annals of Mathematics, 2011, 174(2): 1057-1110.

DOI：10.4007/annals.2011.174.2.7

We introduce a conjugation invariant normalized height (h) over cap (F) on finite subsets of matrices F in GL(d)((Q) over bar) and describe its properties. In particular, we prove an analogue of the Lehmer problem for this height by showing that (h) over cap (F) > epsilon whenever F generates a nonvirtually solvable subgroup of GL(d)((Q) over bar), where epsilon = epsilon(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.

• 出版日期2011-9