  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.
