论文标题
关于杰出组的拓扑生成
On the topological generation of exceptional groups by unipotent elements
论文作者
论文摘要
让$ g $成为一个简单的代数类型类型的代数组，而在代数封闭的特征$ p \ geqslant 0 $上，这不是有限字段的代数。令$ \ Mathcal {C} _1，\ ldots，\ Mathcal {C} _t $是$ g $中的非中央偶联类。在与Gerhardt和Guralnick的早期合作中，我们证明了如果$ t \ geqslant 5 $（或$ t \ geqslant 4 $如果$ g = g_2 $），则存在$ x_i \ in \ nathcal {c} _i $ x_i \ in \ mathcal {c} _i $，这样的$ \ langle x_1， $ g $。此外，这是$ t $的限制。在这里，我们在特殊情况下建立了此结果的更精致版本，其中$ p> 0 $和$ \ mathcal {c} _i $是包含订单$ p $元素的单一类。实际上，在这种情况下，我们完全确定$ \ MATHCAL {C} _1，\ ldots，\ Mathcal {C} _t $ for $ t \ geqslant 2 $ 2 $，使得$ \ langle x_1，\ ldots，x_t \ rangle $ rangle $ is zariski to kariski ins zariski ins zariski ins of zariski to y in $ x__i \ c \ c \ c \ c \ c \ c。
Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p \geqslant 0$ which is not algebraic over a finite field. Let $\mathcal{C}_1, \ldots, \mathcal{C}_t$ be non-central conjugacy classes in $G$. In earlier work with Gerhardt and Guralnick, we proved that if $t \geqslant 5$ (or $t \geqslant 4$ if $G = G_2$), then there exist elements $x_i \in \mathcal{C}_i$ such that $\langle x_1, \ldots, x_t \rangle$ is Zariski dense in $G$. Moreover, this bound on $t$ is best possible. Here we establish a more refined version of this result in the special case where $p>0$ and the $\mathcal{C}_i$ are unipotent classes containing elements of order $p$. Indeed, in this setting we completely determine the classes $\mathcal{C}_1, \ldots, \mathcal{C}_t$ for $t \geqslant 2$ such that $\langle x_1, \ldots, x_t \rangle$ is Zariski dense for some $x_i \in \mathcal{C}_i$.
