Abstract
Let G be a simple algebraic group over an algebraically closed ﬁeld k and let C1,...,Ct be noncentral 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 language  English 

Article number  107177 
Number of pages  50 
Journal  Advances in Mathematics 
Volume  369 
Early online date  7 May 2020 
DOIs  
Publication status  Published  5 Aug 2020 
Keywords
 Topological generation
 Exceptional algebraic groups
 Random generation
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Professor Tim Burness
 School of Mathematics  Professor of Pure Mathematics
 Pure Mathematics
 Algebra
Person: Academic , Member