On the Topological Generation of Exceptional Groups by Unipotent Elements

Abstract

Let G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $$p \geqslant 0$$ p ⩾ 0 which is not algebraic over a finite field. Let $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t be non-central conjugacy classes in G. In earlier work with Gerhardt and Guralnick, we proved that if $$t \geqslant 5$$ t ⩾ 5 (or $$t \geqslant 4$$ t ⩾ 4 if $$G = G_2$$ G = G 2 ), then there exist elements $$x_i \in \mathcal {C}_i$$ x i ∈ C i such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , 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 $$p>0$$ p > 0 and the $$\mathcal {C}_i$$ C i are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t for $$t \geqslant 2$$ t ⩾ 2 such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense for some $$x_i \in \mathcal {C}_i$$ x i ∈ C i .