On the topological generation of exceptional groups by unipotent elements

2022/06/21 by Timothy C. Burness, Burness, Timothy C.
Mathematics · #Finite Group Theory Research #Rings, Modules, and Algebras #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2206.10150

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 C₁, …, 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 = G₂), then there exist elements x_i ∈ C_i such that ⟨ x₁, …, 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 C_i are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes C₁, …, C_t for t \geqslant 2 such that ⟨ x₁, …, x_t ⟩ is Zariski dense for some x_i ∈ C_i.

Related

• Topological generation of exceptional algebraic groups
• Topological generation of simple algebraic groups
• Topological generation of special linear groups
• Generic elements of a Zariski-dense subgroup form an open subset
• The non-Archimedean Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1