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

TL;DR

This paper uses the Fixed Point Theorem for group actions on buildings to analyze algebraic groups over polynomial rings, proving conjugacy properties of finite subgroups and providing a new proof of a torsor theorem.

Contribution

It applies geometric fixed point techniques to establish conjugacy results for finite subgroups and offers a simplified proof of a classical torsor theorem over the affine line.

Findings

Finite subgroups of G(k[t]) are conjugate to subgroups of G(k).

G(k[t]) has finitely many conjugacy classes of finite subgroups over certain fields.

Provides a short proof of the Raghunathan-Ramanathan theorem on G-torsors.

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.