Topological generation of simple algebraic groups

TL;DR

This paper investigates conditions under which tuples of elements in simple algebraic groups generate Zariski dense subgroups, providing a comprehensive solution for symplectic, orthogonal, linear, and exceptional groups, with applications to representation theory and probabilistic generation.

Contribution

It offers a complete characterization of generically dense tuples in simple algebraic groups, especially for conjugacy classes of prime order, extending previous results to all types of simple groups.

Findings

Complete solutions for symplectic and orthogonal groups.

Applications to faithful representations being generically free.

Asymptotic results on probabilistic generation of finite simple groups.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1, \ldots, x_r) \in X$ such that $\langle x_1, \ldots, x_r \rangle$ is Zariski dense in $G$. We are primarily interested in the case where $X = C_1 \times \cdots \times C_r$ and each $C_i$ is a conjugacy class of $G$ comprising elements of prime order modulo the center of $G$. In this setting, our main theorem gives a complete solution to the problem when $G$ is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives a complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.