例外代数群的拓扑生成
Topological generation of exceptional algebraic groups
摘要：
设$G$是代数闭域$k$上的单代数群，$C_1, \ldots, C_t$是$G$中的非中心共轭类.本文考虑了在$G$中是否存在$g_i \in C_i$使得$\langle g_1, \ldots, g_t \rangle$是Zerkiki稠密的问题.首先我们建立了一个一般性的结果，证明了如果$Ω$是$G^t$的一个不可约子簇，则生成$G$的稠密子群的元组集在$Ω$中要么是空的，要么是稠密的.在特殊情形$Ω= C_1 \times \cdots \times C_t$下，通过考虑不动点空间的维数，证明了当$G$是例外代数群，且$t \geqslant 5$（假定$k$不是有限域上的代数群）是例外代数群时，该不动点集是稠密的.事实上，对于$G=G_2$，我们只需要$t \geqslant 4$，并且这两个边界都是最佳可能的。作为应用，我们证明了例外代数群的许多忠实表示是泛自由的。我们还建立了特殊情况下$t=2$例外群的拓扑生成的新结果，这些结果在李型有限例外群的随机生成中有应用。特别地，我们证明了Liebeck和Shalev关于例外群的随机$(r,s)$ -生成的一个猜想.
英文摘要：
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 $Ω$ is an irreducible subvariety of $G^t$, then the set of tuples in $Ω$ generating a dense subgroup of $G$ is either empty or dense in $Ω$. In the special case $Ω= C_1 \times \cdots \times C_t$, 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.
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