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 ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_t$ be non-central conjugacy classes in $G$. In earlier work with Gerhardt and Guralnick, we proved that if $t⩾5$ (or $t⩾4$ if $G=G_2$), then there exist elements $x_i\in {\mathcal{C}}_i$ such that $⟨x_1,\dots ,x_t⟩$ 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 ${\mathcal{C}}_i$ are unipotent classes containing elements of order $p$. Indeed, in this setting we completely determine the classes ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_t$ for $t⩾2$ such that $⟨x_1,\dots ,x_t⟩$ is Zariski dense for some $x_i\in {\mathcal{C}}_i$.