Group actions and power maps for groups over non-Archimedean local fields

2021/03/11 by Arunava Mandal, Mandal, Arunava, C. R. E. Raja +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2103.06612

openalex publication_date 2021/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider linear groups and Lie groups over a non-Archimedean local field \mathbb F for which the power map x↦ x^k has a dense image or it is surjective. We prove that the group of \mathbb F-points of such algebraic groups is a compact extension of unipotent groups with the order of the compact group being relatively prime to k. This in particular shows that the power map is surjective for all k is possible only when the group is unipotent or trivial depending on whether the characteristic of \mathbb F is zero or positive. Similar results are proved for Lie groups via the adjoint representation. To a large extent, these results are extended to linear groups over local fields and global fields.

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