2022/03/29 by Glockner, Helge
#22E20 (Primary) 22E35 #22E50 #32P05 (Secondary) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2203.15861
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→ G with f(M)⊆ 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.