On the Nilpotency of Locally Pro-p Contraction Groups

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 ${\mathbb{F}}_p(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$ acts on its $d$-th power ${\mathbb{F}}_p(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣}{)}^d$ 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.