Non-Lie Subgroups in Lie groups over Local Fields of Positive Characteristic

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 $H$.