Topological generation of exceptional algebraic groups

Tim C Burness*, Spencer Gerhardt*, Robert Guralnick*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

Let G be a simple algebraic group over an algebraically closed field k and let C1,...,Ct be non-central conjugacy classes in G. In this paper, we consider the problem of determining whether there exist gi ∈ Ci such that hg1,...,gti is Zariski dense in G. First we establish a general result, which shows that if Ω is an irreducible subvariety of Gt, then the set of tuples in Ω generating a dense subgroup of G is either empty or dense in Ω. In the special case Ω = C1 ×···× Ct, by considering the dimensions of fixed point spaces, we prove that this set is dense when G is an exceptional algebraic group and t > 5, assuming k is not algebraic over a finite field. In fact, for G = G2 we only need t > 4 and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case t = 2, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random (r,s)-generation of exceptional groups.
Original languageEnglish
Article number107177
Number of pages50
JournalAdvances in Mathematics
Volume369
Early online date7 May 2020
DOIs
Publication statusPublished - 5 Aug 2020

Keywords

  • Topological generation
  • Exceptional algebraic groups
  • Random generation

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