On the Nilpotency of Locally Pro-p Contraction Groups

TL;DR

This paper provides a concise proof that locally pro-p contraction groups are nilpotent, building on a fixed-point theorem related to actions on local fields, which simplifies previous arguments.

Contribution

It offers a shorter proof of the nilpotency of locally pro-p contraction groups by establishing a fixed-point result for actions on local fields.

Findings

Locally pro-p contraction groups are nilpotent.

A fixed-point theorem for actions on local fields is proven.

Simplified proof of a key result in the theory of contraction groups.

Abstract

H. Gl\"ockner and G. A. Willis have recently shown that locally pro-p contraction groups are nilpotent. The proof hinges on a fixed-point result: if the local field $\mathbb{F}_{p}(\!(t)\!)$ acts on its $d$-th power $\mathbb{F}_{p}(\!(t)\!)^{d}$ additively, continuously, and in an appropriately equivariant manner, then the action has a non-zero fixed point. We provide a short proof of this theorem.