  We show that finitely generated subhomogeneous C*-algebras have finite
decomposition rank. As a consequence, any separable ASH C*-algebra can be
written as an inductive limit of subhomogeneous C*-algebras each of which has
finite decomposition rank. It then follows from work of H. Lin and of the
second named author that the class of simple unital ASH algebras which have
real rank zero and absorb the Jiang-Su algebra tensorially satisfies the
Elliott conjecture.
