论文标题
固定点定理的应用用于对多项式环的代数组的建筑物的组动作
Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings
论文作者
论文摘要
我们将固定点定理应用于有限组对Bruhat-tits建筑物及其产品的作用，以在一个变量中在一个变量的多项式环上建立有关还原代数组的两个结果的两个结果，因为假设基础场是特征性零的。首先，我们证明，对于还原$ k $ -group $ g $，每个有限亚组$ g（k [t]）$与$ g（k）$的子组相连。尤其是这意味着，如果$ k $是$ p $ - ad的字段$ \ mathbb {q} _p $的有限扩展，那么组$ g（k [t]）$有限许多有限的子组类别有限，这是算术组众所周知的财产。其次，我们简短地证明了raghunathan-ramanathan定理，大约$ g $ tostors offine line。
We apply the Fixed Point Theorem for the actions of finite groups on Bruhat-Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive $k$-group $G$, every finite subgroup of $G(k[t])$ is conjugate to a subgroup of $G(k)$. This, in particular, implies that if $k$ is a finite extension of the $p$-adic field $\mathbb{Q}_p$, then the group $G(k[t])$ has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a give a short proof of the theorem of Raghunathan-Ramanathan about $G$-torsors over the affine line.
