Strongly dense free subgroups of semisimple algebraic groups

TL;DR

This paper proves the existence of strongly dense free subgroups in semisimple algebraic groups over large fields, with implications for finite simple groups, Borel's theorem, and expansion properties of Cayley graphs.

Contribution

It establishes the existence of strongly dense free subgroups in semisimple algebraic groups, enhancing understanding of their structure and applications.

Findings

Existence of strongly dense free subgroups in most semisimple algebraic groups

New generating results for finite simple groups of Lie type

Strengthening of Borel's theorem related to the Hausdorff-Banach-Tarski paradox

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.