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 ∈ Ci such that hx1, . . . , xti 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 hx1, . . . , xti is Zariski dense for some xi ∈ Ci.