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

TL;DR

This paper investigates the surjectivity and density of power maps in linear and Lie groups over non-Archimedean local fields, revealing structural conditions under which these maps are surjective or dense.

Contribution

It establishes the structure of groups with dense or surjective power maps over non-Archimedean fields, showing they are compact extensions of unipotent groups with specific properties.

Findings

Power map surjectivity implies the group is unipotent or trivial.

Groups with dense power maps are compact extensions of unipotent groups.

Results extend to linear groups over local and global fields.

Abstract

We consider linear groups and Lie groups over a non-Archimedean local field $\mathbb F$ for which the power map $x\mapsto x^k$ 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.