Abelian maximal subgroups of valued division algebras

Abstract

Let $D$ be a division ring. We study when the multiplicative group $D^{\ast }$ can contain an abelian maximal subgroup. We prove that this cannot occur for any noncommutative tame finite-dimensional central division algebra over a Henselian valued field with nontrivial valuation. We also apply the same idea to standard valued division rings, including twisted Laurent series, iterated Laurent series, and Mal'cev--Neumann series. Finally, we introduce a complementary malnormality obstruction and use it to rule out abelian maximal subgroups in quaternion division algebras over real-closed fields.