derived length bounds for soluble-by-finite subgroups in valued division algebras

We study soluble-by-finite subgroups of the multiplicative group of a valued division algebra. when the residue division algebra is a field and the valuation is strongly tame, every soluble-by-finite subgroup of the multiplicative group is soluble with derived length at most three. in two important cases (totally ramified algebras and prime powerdegree algebras without a primitive root of unity in the center) the bound improves to two. several examples constructed via iterated laurent-series and symbol algebras illustrate the necessity of the hypotheses.