Strongly dense free subgroups of semisimple algebraic groups

2010/10/20 by Breuillard, Emmanuel, Green, Ben, Guralnick, Robert +1 · 1 citation
#20G40 #20N99 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1010.4259

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.

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• Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks

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