On the Nilpotency of Locally Pro-p Contraction Groups
Alonso Beaumont
TL;DR
The paper addresses the fixed-point problem underlying the nilpotency of locally pro-$p$ contraction groups by examining a continuous, equivariant action of $G=\mathbb{F}_{p}((t))$ on $\mathbb{F}_{p}((t))^{d}$. It introduces a two-step, representation-theoretic approach augmented by Ascoli's theorem to construct invariant subspaces and ultimately locate a non-zero fixed point, which yields the desired nilpotency result. The main contribution is a concise alternative to prior endomorphism-matrix arguments, strengthening the fixed-point foundation for the theory of contraction groups in the locally pro-$p$ setting. This streamlined argument enhances understanding of the torsion structure and supports broader structural results for tdloc contraction groups.
Abstract
H. Glöckner 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.