## 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*⩾2. In this paper, we consider the general problem of determining if there exists a tuple (*x*_{1},…,*x*_{r})∈*X*such that ⟨*x*_{1},…,*x*⟩ is Zariski dense in_{r}*G*. We are primarily interested in the case where*X*=*C*_{1}×⋯×*C*and each_{r }*C*is a conjugacy class of_{i }*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 an almost 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.Original language | English |
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Number of pages | 69 |

Journal | Journal of the European Mathematical Society |

Early online date | 8 Apr 2024 |

DOIs | |

Publication status | E-pub ahead of print - 8 Apr 2024 |