Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings

Abstract: 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.