Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, \ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i \in C_i$ such that $\langle g_1, \ldots, g_t \rangle$ is Zariski dense in $G$. First we establish a general result, which shows that if $\Omega$ is an irreducible subvariety of $G^t$, then the set of tuples in $\Omega$ generating a dense subgroup of $G$ is either empty or dense in $\Omega$. In the special case $\Omega = C_1 \times \cdots \times C_t$, by considering the dimensions of fixed point spaces, we prove that this set is dense when $G$ is an exceptional algebraic group and $t \geqslant 5$, assuming $k$ is not algebraic over a finite field. In fact, for $G=G_2$ we only need $t \geqslant 4$ and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case $t=2$, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random $(r,s)$-generation of exceptional groups.

中文翻译：

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异常代数群的拓扑生成

令 $G$ 是代数闭域 $k$ 上的一个简单代数群，令 $C_1, \ldots, C_t$ 是 $G$ 中的非中心共轭类。在本文中，我们考虑确定是否存在$g_i \in C_i$ 使得$\langle g_1, \ldots, g_t \rangle$ 在$G$ 中是Zariski 稠密的问题。首先我们建立一个一般结果，它表明如果 $\Omega$ 是 $G^t$ 的一个不可约子变体，那么 $\Omega$ 中生成 $G$ 的稠密子群的元组集要么是空的，要么是稠密的$\欧米茄$。在特殊情况 $\Omega = C_1 \times \cdots \times C_t$，通过考虑不动点空间的维数，我们证明当 $G$ 是一个例外代数群并且 $t \geqslant 5$ 时，这个集合是稠密的，假设 $k$ 在有限域上不是代数的。实际上，对于 $G=G_2$ 我们只需要 $t \geqslant 4$ 并且这两个边界都是最好的。作为一个应用，我们证明了异常代数群的许多忠实表示是一般自由的。我们还在特殊情况 $t=2$ 中建立了异常群拓扑生成的新结果，该结果可应用于 Lie 类型有限异常群的随机生成。特别是，我们证明了 Liebeck 和 Shalev 关于随机 $(r,s)$-生成异常群的猜想。