Comptes Rendus
Research article - Group theory
On the nilpotency of locally pro-$p$ contraction groups
Alonso Beaumont1
1 IRMAR, Université de Rennes, 263 Avenue du Général Leclerc, 35042 Rennes Cedex, France
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 267-270.

H. Glöckner and G. A. Willis have recently shown [2] that locally pro-$p$ contraction groups are nilpotent. The proof hinges on a fixed point result [2, Theorem B]: 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.

H. Glöckner et G. A. Willis ont récemment démontré [2] que les groupes de contraction localement pro-$p$ sont nilpotents. Leur démonstration repose sur un résultat de point fixe [2, Theorem B] : si le corps local $\mathbb{F}_{p}(\!(t)\!)$ agit sur sa $d$-ième puissance $\mathbb{F}_{p}(\!(t)\!)^{d}$ additivement, continûment et de manière convenablement équivariante, alors l’action a un point fixe non nul. Nous présentons une démonstration courte de ce théorème.

Received:
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DOI: 10.5802/crmath.728
Classification: 22D05, 20E36, 20F18

Alonso Beaumont 1

1 IRMAR, Université de Rennes, 263 Avenue du Général Leclerc, 35042 Rennes Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {On the nilpotency of locally pro-$p$ contraction groups},
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Alonso Beaumont. On the nilpotency of locally pro-$p$ contraction groups. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 267-270. doi : 10.5802/crmath.728. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.728/

[1] Helge Glöckner; George A. Willis Classification of the simple factors appearing in composition series of totally disconnected contraction groups, J. Reine Angew. Math., Volume 2010 (2010) no. 643, pp. 141-169 | MR | Zbl

[2] Helge Glöckner; George A. Willis Locally pro-p contraction groups are nilpotent, J. Reine Angew. Math., Volume 781 (2021), pp. 85-103 | DOI | MR | Zbl

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[6] Markus Stroppel Locally compact groups, EMS Textbooks in Mathematics, European Mathematical Society, 2006, x+302 pages | DOI | MR

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