The generator rank of subhomogeneous C*-algebras

We compute the generator rank of a subhomgeneous C*-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every Z-stable ASH-algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear C*-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear C*-algebras.