On the Topological Generation of Exceptional Groups by Unipotent Elements

Let G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p >= 0 which is not algebraic over a finite field. Let C1,..., Ct be non-central conjugacy classes in G. In earlier work with Gerhardt and Guralnick, we proved that if t >= 5 (or t >= 4 if G = G2), then there exist elements xi is an element of C-i such that < x1,..., xt > is Zariski dense in G. Moreover, this bound on t is best possible. Here we establish a more refined version of this result in the special case where p > 0 and the Ci are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes C1,..., Ct for t >= 2 such that < x1,..., xt > is Zariski dense for some xi is an element of C-i