On the Nilpotency of Locally Pro-p 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.