<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%~MoulinÃ© par MaN_auto v.0.33.5 2025-03-18 14:56:21
\documentclass[CRMATH,Unicode,XML,biblatex]{cedram}

\TopicFR{ThÃ©orie des groupes}
\TopicEN{Group theory}

\usepackage{mathtools}

\addbibresource{CRMATH_Beaumont_20250022.bib}

\newcommand{\Q}{\mathbb{Q}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Fpt}{\F_{p}(\!(t)\!)}
\newcommand{\Fptt}{\F_{p}\llbracket t\rrbracket}

\DeclareMathOperator{\Aut}{Aut}
\newcommand*{\id}{\mathrm{id}}

%%%---------------------------------------------------------------------------

\newcounter{step}
\newenvironment{step}[1][]{\refstepcounter{step}\begin{proof}[{Step~\thestep\if#1\empty\else: #1\fi}]}{\let\qed\relax\end{proof}}


%%%---------------------------------------------------------------------------

%%% Changer la largeur des adresses
% Jouer avec \hsize :
\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=11cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

%%%---------------------------------------------------------------------------

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\let\oldforall\forall
\renewcommand*{\forall}{\mathrel{\oldforall}}

\title{On the nilpotency of locally pro-$p$ contraction groups}
\alttitle{Sur la nilpotence des groupes de contraction localement pro-$p$}

\author{\firstname{Alonso} \lastname{Beaumont}}
\address{IRMAR, UniversitÃ© de Rennes, 263 Avenue du GÃ©nÃ©ral Leclerc, 35042 Rennes Cedex, France}
\email{alonso.beaumont@univ-rennes.fr}

\subjclass{22D05, 20E36, 20F18}

\begin{abstract}
H.\ GlÃ¶ckner and G.\ A.\ Willis have recently shown~\cite{GW21} that locally pro-$p$ contraction groups are nilpotent. The proof hinges on a fixed point result~\cite[Theorem~B]{GW21}: 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.
\end{abstract}

\begin{altabstract}
H.\ GlÃ¶ckner et G.\ A.\ Willis ont rÃ©cemment dÃ©montrÃ©~\cite{GW21} que les groupes de contraction localement pro-$p$ sont nilpotents. Leur dÃ©monstration repose sur un rÃ©sultat de point fixe~\cite[Theorem~B]{GW21}: si le corps local $\mathbb{F}_{p}(\!(t)\!)$ agit sur sa $d$-iÃ¨me puissance $\mathbb{F}_{p}(\!(t)\!)^{d}$ additivement, continÃ»ment et de maniÃ¨re convenablement Ã©quivariante, alors l'action a un point fixe non nul. Nous prÃ©sentons une dÃ©monstration courte de ce thÃ©orÃ¨me.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

A locally compact topological group $G$ is said to be a \emph{contraction group} if it has a continuous automorphism $\alpha$ whose forward orbits converge to the identity element $e$:
\[
\forall g\in G, \quad \alpha^{n}(g)\underset{n\rightarrow\infty}{\longrightarrow} e.
\]
This dynamical property has interesting connections with the purely group-theoretic notion of nilpotency. This relationship is best illustrated in the case where $G$ is a Lie group: a contracting automorphism on $G$ provides a positive gradation for its Lie algebra $\mathfrak{g}$, proving in particular that $\mathfrak{g}$ is nilpotent (see~\cite{S86}). A thorough study of totally disconnected locally compact (tdlc) contraction groups is carried out in~\cite{GW10}. One of its main structural results asserts that any tdlc contraction group is a direct product of a torsion-free contraction group, which is always nilpotent, and a torsion contraction group~\cite[Theorem~B]{GW10}. The remaining questions concerning the torsion part were treated in~\cite{GW21}. In order to accommodate for torsion, an additional hypothesis is needed: a tdlc group is called \emph{locally pro-$p$} if it contains an open pro-$p$ subgroup. For example, analytic groups over the field $\Q_{p}$ of $p$-adic numbers, or the field $\Fpt$ of Laurent series with coefficients in $\F_{p}$, are locally pro-$p$. In~\cite{GW21} it is shown that locally pro-$p$ contraction groups are nilpotent. The major part of the proof consists in establishing the following fixed point result~\cite[Theorem~B]{GW21}:

\begin{theorem}\label{theo}
Let $d\geq1$ and consider a continuous action $\psi\colon\bigl(\Fpt,+\bigr)\times \bigl(\Fpt^{d},+\bigr)\rightarrow \bigl(\Fpt^{d},+\bigr)$ by group automorphisms. If
\[
\forall x\in\Fpt,\;\forall u\in \Fpt^{d},\; \psi(tx,tu)=t\psi(x,u)
\]
then the action has a non-zero fixed point.
\end{theorem}

GlÃ¶ckner and Willis's proof of this result involves representing endomorphisms of the abelian group $\Fpt^d$ as infinite block matrices satisfying certain conditions. We refer the reader to Section~3 of~\cite{GW21} for details. The purpose of this note is to provide an alternative, shorter proof of this result. It has two main ingredients: a study of $\F_{p}$-representations of elementary abelian $p$-groups, and an application of Ascoli's theorem to the automorphism group $\Aut\bigl(\Fpt^{d}\bigr)$.

\section{Proof of the theorem}

We begin with a lemma. A linear representation $V$ of a group $G$ is said to be \emph{locally finite} if its finitely generated subrepresentations are finite-dimensional. We denote by $V^{G}$ the set of $G$-fixed points in $V$.

\begin{lemma}\label{lem}
Let $G$ be an elementary abelian $p$-group, and $V$ a locally finite representation of\/ $G$ over $\F_{p}$. If\/ $V$ is infinite-dimensional, then either
\[
V^{G} \quad or \quad \bigl(V/V^{G}\bigr)^{G}
\]
is infinite-dimensional.
\end{lemma}

\begin{proof}
First, suppose that $G$ is finite. We will show that $V^{G}$ is infinite-dimensional. In order to give a self-contained presentation, we will provide a direct proof of the lemma in this case\footnote{An alternative line of argument is as follows. If $G$ is finite then $\F_{p}[G]$ is a finite-dimensional $\F_{p}$-algebra. If the $\F_{p}[G]$-module $V^{G}$ were finite then its essential extension $V$ would also be finite (see for instance~\cite{RZ58}), so $V^{G}$ must be infinite-dimensional.}. Write $G=\langle g_{1},\dotsc, g_{r}\rangle$. By the local finiteness of $V$, there is a strictly increasing family $(V_{n})_{n\geq0}$ of finite subrepresentations. For all $n\geq0$, write $d_{n}=\dim V_{n}$ and consider the sequence
\[
\forall i\in\{0,\dotsc, p\},\quad d_{n}(i)=\dim\biggl(\ker(g_{1}-\id)^{i}\cap V_{n}\biggr).
\]
By a standard application of the rank-nullity theorem, $(d_{n}(i+1)-d_{n}(i))_{0\leq i&lt; p}$ is a decreasing sequence. Moreover, $d_{n}(p)=d_{n}$ since $(g_{1}-\id)^{p}=g_{1}^{p}-\id=0$. We deduce that $d_{n}(1)=\dim V_{n}^{\langle g_{1}\rangle}\geq d_{n}/p$. The elements $g_{1}$ and $g_{2}$ commute, so we can consider the action of $\langle g_{2}\rangle$ on $V^{\langle g_{1}\rangle}$. By repeating the argument above we obtain $\dim V_{n}^{\langle g_{1},g_{2}\rangle}\geq d_{n}/p^{2}$ and by a finite induction,
\[
\dim V^{G}\geq\dim V_{n}^{G}\geq d_{n}/p^{r}\underset{n\rightarrow+\infty}{\longrightarrow}+\infty.
\]

Now suppose that $G$ is infinite, and that both $V^{G}$ and $(V/V^{G})^{G}$ are finite-dimensional. Then the preimage $V'$ of $(V/V^{G})^{G}$ in $V$ is finite-dimensional. Therefore, the $G$-action on $V'$ factors through a finite quotient of $G$. Since $G$ is elementary abelian, this quotient is a direct summand. We can therefore write $G=G_{0}\times G_{1}$, where $G_{0}$ is finite and $G_{1}$ acts trivially on $V'$. In particular, $G_{1}$ acts trivially on $V'\cap V^{G_{0}}$ so $V'\cap V^{G_{0}}\subset (V^{G_{0}})^{G_{1}}=V^{G}$. Hence,
\[
\bigl(V'\cap V^{G_{0}}\bigr)/V^{G}=\bigl(V^{G_{0}}/V^{G}\bigr)^{G}=0.
\]
Suppose $V^{G_{0}}/V^{G}$ is non-trivial. Since $V^{G_{0}}/V^{G}$ is locally finite, it would have a non-trivial finite-dimensional subrepresentation, and hence a non-zero $G$-fixed point, because $G$ acts as an abelian unipotent group. We deduce that $V^{G_{0}}/V^{G}=0$. This is absurd since $G_{0}$ is finite and thus $V^{G_{0}}$ is infinite-dimensional.
\end{proof}

Let $d\geq1$. We consider $\Fpt^{d}$ as an additive tdlc group, equipped with a continuous automorphism given by scalar multiplication by $t$: $u=(u_{1},\dotsc,u_{d})\mapsto tu=(tu_{1},\dotsc,tu_{d})$. The group $\Aut\bigl(\Fpt^{d}\bigr)$ of continuous automorphisms is endowed with the compact-open topology. In particular, a continuous action $G\times\Fpt^{d}\rightarrow\Fpt^{d}$ by group automorphisms induces a continuous morphism $\varphi\colon G\rightarrow\Aut\bigl(\Fpt^{d}\bigr)$. We restate Theorem~\ref{theo} in the following manner:

\begin{theorem}\label{theor}
A continuous morphism $\varphi\colon\Fpt\rightarrow\Aut\bigl(\Fpt^{d}\bigr)$ satisfying
\begin{equation}\label{1}
\forall x\in\Fpt,\;\forall u\in \Fpt^{d},\;\varphi(tx)(u)=t\varphi(x)(t^{-1}u)
\end{equation}
has a non-zero fixed point: there exists $u\in\Fpt^{d}\backslash\{0\}$ such that $\varphi(x)(u)=u$ for all $x\in\Fpt$.
\end{theorem}

\begin{proof}
The proof will be done in two steps. We will first construct a nested family of invariant subgroups, and then we will use it to locate increasingly smaller orbits that will converge to a fixed point. We will use the following notation: $G=\varphi\bigl(\Fpt\bigr)$, $G_{\ell}=\varphi\bigl(t^{-\ell}\Fptt\bigr)$ for all $\ell\geq0$, $B=\Fptt^{d}$, and $S=B\backslash tB$.

\begin{step}\label{step_1}
Let $M$ be the largest $G$-invariant subgroup contained in $B$. It can be defined as the group generated by all $G$-invariant subgroups in $B$. We will prove the following: $M$ is non-trivial and closed, and $A\coloneqq \bigcup_{n\geq0}t^{-n}M$ is a $G$-invariant $\Fpt$-subspace of $\Fpt^{d}$ in which $M$ is open.

Since $\varphi$ is continuous, $G_{\ell}$ is compact, and so by Ascoli's theorem~\cite[Theorem~47.1]{M14} it is an equicontinuous family of automorphisms of $\Fpt^{d}$. In particular, there exists a neighbourhood of $0$ whose $G_{\ell}$-orbit is contained in $B$. The subgroup generated by this orbit is therefore open, $G_{\ell}$-invariant, and contained in $B$. Let $M_{\ell}$ be the largest $G_{\ell}$-invariant subgroup in $B$. By the previous argument $M_{\ell}$ contains an open subgroup and thus it is itself open. By property~\eqref{1}, we have
\[
\forall x\in t^{-\ell}\Fptt,\quad \varphi(x)(t^{-1}M_{\ell})=t^{-1}\varphi(tx)(M_{\ell})=t^{-1}M_{\ell}
\]
and so $t^{-1}M_{\ell}$ is also $G_{\ell}$-invariant. This means that $M_{\ell}$ and $S$ have a non-empty intersection, for otherwise the subgroup $M_{\ell}+t^{-1}M_{\ell}$, which is $G_{\ell}$-invariant and strictly larger that $M_{\ell}$, would be contained in $B$. Therefore, the decreasing sequence $(M_{\ell}\cap S)_{\ell\geq0}$ consists of non-empty compact sets, and hence its intersection is non-empty. We have proven that
\[
M\coloneqq \bigcap_{\ell\geq0}M_{\ell}
\]
is a non-trivial subgroup of $\Fpt^{d}$. Every $G$-invariant subgroup of $B$ is $G_{\ell}$-invariant, hence contained in $M_{\ell}$ for each $\ell$. Therefore $M$ is the largest $G$-invariant subgroup of $B$. Again using~\eqref{1}, we see that $tM$ is $G$-invariant and so $tM\subset M$. As an intersection of closed subgroups, $M$ is closed. We deduce that $M$ is an $\Fptt$-module and thus
\[
A\coloneqq \bigcup_{n\geq0}t^{-n}M
\]
is a $G$-invariant $\Fpt$-subspace of $\Fpt^{d}$. In particular, $A$ is closed in $\Fpt^{d}$ and therefore it is a Baire space. As such, some $t^{-n}M$, or equivalently all of them, are open in $A$.
\end{step}

\begin{step} For all $n\geq0$, let $V_{n}\coloneqq t^{-n}M/M$. This quotient is an $\F_{p}$-vector space of finite dimension endowed with a linear $G$-action. This means that $V\coloneqq A/M=\bigcup_{n\geq0}V_{n}$ satisfies the hypotheses of Lemma~\ref{lem}. Let us show that $V$ has infinitely many $G$-fixed points. Suppose otherwise: $V^{G}$ would then be contained in $V_{n}$ for some $n\geq0$. On the other hand, by Lemma~\ref{lem} $(V/V^{G})^{G}$ would be infinite: since $V/V^{G}$ surjects $G$-equivariantly onto $V/V_{n}$ with finite kernel, $(V/V_{n})^{G}$ would also be infinite. We claim this is absurd because $V^{G}$ and $(V/V_{n})^{G}$ are in bijection with one another. Indeed, we may choose a section $V/V_{n}=A/t^{-n}M\rightarrow A$ and compose it with $t^{n}\cdot\id\colon A\rightarrow A$ followed by $A\rightarrow V$. This defines an isomorphism $V/V_{n}\rightarrow V$ which by property~\eqref{1} preserves $G$-fixed points.

We have shown that $V^{G}$ is infinite. In particular, for all $n\geq0$, $V^{G}$ is not contained in $V_{n}$: we can find some $\overline{v_{n}}\in (A/M)^{G}$ that lifts to $v_{n}\in A\backslash t^{-n}M$. Let $k_{n}&gt; n$ be the unique integer such that $w_{n}\coloneqq t^{k_{n}}v_{n}\in M\backslash tM$. Applying~\eqref{1}, we have
\[
\forall x\in \Fpt,\quad \varphi(x)(w_{n})-w_{n}=t^{k_{n}}\bigl(\varphi(t^{-k_{n}}x)(v_{n})-v_{n}\bigr)\in t^{k_{n}}M.
\]
Since $M\backslash tM$ is compact, the family $(w_{n})_{n\geq0}$ has an accumulation point $w$. For all $x\in\Fpt$ and infinitely many $n$, we have $\varphi(x)(w)-w\in t^{k_{n}}M$, so $\varphi(x)(w)=w$: we have constructed a non-zero $G$-fixed point in $A$.\qedhere
\end{step}
\let\qed\relax
\end{proof}

\begin{remark}
GlÃ¶ckner and Willis state Theorem~\ref{theor} slightly differently: they endow $\Aut\bigl(\Fpt^{d}\bigr)$ with the finer Braconnier topology, and assume that $\varphi$ is continuous with respect to it. This assumption is in fact no stronger than ours (see~\cite[Lemma~9.13]{S06}). In light of this, Step~\ref{step_1} of the proof above may be started as follows. The subgroup $\bigl\{g\in G_{\ell}\,\big\vert\, g(B)=B\bigr\}$ is open in $G_{\ell}$ for the Braconnier topology, and therefore it is of finite index. Thus, the intersection $\bigcap_{g\in G_{\ell}}g(B)$ is finite, yielding an open $G_{\ell}$-invariant subgroup in $B$.
\end{remark}

\begin{remark}
Let $(G_{1},\alpha_{1})$ and $(G_{2},\alpha_{2})$ be two tdlc contraction groups, equipped with their respective contracting automorphisms. A continuous action $\psi\colon G_{1}\times G_{2}\rightarrow G_{2}$ may be called equivariant if $\psi\bigl(\alpha_{1}(x),\alpha_{2}(y)\bigr)=\alpha_{2}\bigl(\psi(x,y)\bigr)$ for all $x\in G_{1}$, $y\in G_{2}$. Then the argument of Step~\ref{step_1} can be applied in this more general setting: there exists a compact, non-trivial subgroup $H$ of $G_{2}$ such that $\alpha_{2}(H)\subset H$ and $\psi(G_{1}\times H)=H$.
\end{remark}

\section*{Acknowledgments}

I'd like to thank David J.\ Benson for his contribution to Lemma~\ref{lem}, as well as Adrien Le Boudec and Helge GlÃ¶ckner for their helpful comments.

\section*{Declaration of interests}

The author does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond his research organization.

\printbibliography
\end{document}</pre></body></html>