On the topological generation of exceptional groups by unipotent elements

TL;DR

This paper refines previous results on generating exceptional algebraic groups with elements from conjugacy classes, focusing on unipotent classes in positive characteristic and characterizing when such elements generate Zariski dense subgroups.

Contribution

It provides a complete classification of unipotent conjugacy classes in exceptional groups that generate Zariski dense subgroups when combined.

Findings

Determines classes that generate Zariski dense subgroups for t ≥ 2

Focuses on unipotent classes with elements of order p in characteristic p>0

Refines earlier bounds on the number of classes needed for density

Abstract

Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p \geqslant 0$ which is not algebraic over a finite field. Let $\mathcal{C}_1, \ldots, \mathcal{C}_t$ be non-central conjugacy classes in $G$. In earlier work with Gerhardt and Guralnick, we proved that if $t \geqslant 5$ (or $t \geqslant 4$ if $G = G_2$), then there exist elements $x_i \in \mathcal{C}_i$ such that $\langle x_1, \ldots, x_t \rangle$ 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 $\mathcal{C}_1, \ldots, \mathcal{C}_t$ for $t \geqslant 2$ such that $\langle x_1, \ldots, x_t \rangle$ is Zariski dense for some $x_i \in \mathcal{C}_i$.