Strongly dense free subgroups of semisimple algebraic groups
Abstract
We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type.
Cite
@article{arxiv.1010.4259,
title = {Strongly dense free subgroups of semisimple algebraic groups},
author = {Emmanuel Breuillard and Ben Green and Robert Guralnick and Terence Tao},
journal= {arXiv preprint arXiv:1010.4259},
year = {2011}
}
Comments
27 pages, no figures, submitted, Israel J. Math. It turns out that there is one specific family of algebraic group - namely Sp(4) in characteristic 3 - which our methods are unable to resolve, and we have adjusted the paper to exclude this case from the results