Non-Lie subgroups in Lie groups over local fields of positive characteristic

TL;DR

This paper constructs examples of non-Lie subgroups within Lie groups over local fields of positive characteristic, showing these subgroups lack compatible analytic manifold structures, contrasting classical Lie group theory over real or p-adic fields.

Contribution

It demonstrates the existence of non-discrete, compact subgroups in such Lie groups that cannot be endowed with compatible analytic manifold or Lie group structures.

Findings

Existence of non-Lie subgroups in positive characteristic Lie groups

These subgroups are compact, non-discrete, and analytic maps into them are locally constant

Such subgroups lack compatible analytic manifold and Lie group structures

Abstract

By Cartan's Theorem, every closed subgroup $H$ of a real (or $p$-adic) Lie group $G$ is a Lie subgroup. For Lie groups over a local field ${\mathbb K}$ of positive characteristic, the analogous conclusion is known to be wrong. We show more: There exists a ${\mathbb K}$-analytic Lie group $G$ and a non-discrete, compact subgroup $H$ such that, for every ${\mathbb K}$-analytic manifold $M$, every ${\mathbb K}$-analytic map $f\colon M\to G$ with $f(M)\subseteq H$ is locally constant. In particular, the set $H$ does not admit a non-discrete ${\mathbb K}$-analytic manifold structure which makes the inclusion of $H$ into $G$ a ${\mathbb K}$-analytic map. We can achieve that, moreover, $H$ does not admit a ${\mathbb K}$-analytic Lie group structure compatible with the topological group structure induced by $G$ on $H$.