Topological generation of special linear groups

TL;DR

This paper characterizes when tuples from specific conjugacy classes in special linear groups generate Zariski dense subgroups, advancing understanding of algebraic group generation, stabilizers, and finite group generation.

Contribution

It provides necessary and sufficient conditions for tuples to generate Zariski dense subgroups in SL_n(k), linking conjugacy classes to group generation properties.

Findings

Conditions for Zariski dense generation in SL_n(k)

New results on generic stabilizers in linear representations

Strengthened results on random (r,s)-generation of finite groups of Lie type

Abstract

Let $C_1,\ldots,C_e$ be noncentral conjugacy classes of the algebraic group $G=SL_n(k)$ defined over a sufficiently large field $k$, and let $\Omega:=C_1\times \ldots \times C_e$. This paper determines necessary and sufficient conditions for the existence of a tuple $(x_1,\ldots,x_e)\in\Omega$ such that $\langle x_1,\ldots,x_e\rangle$ is Zariski dense in $G$. As a consequence, a new result concerning generic stabilizers in linear representations of algebraic groups is proved, and existing results on random $(r,s)$-generation of finite groups of Lie type are strengthened.