  Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension
growth. We prove that if A is unital and simple, then the Cuntz semigroup of A
agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem
with a result of W. Winter, this yields the equivalence of Z-stability and slow
dimension growth for unital simple ASH algebras. This equivalence has several
consequences, including the following classification theorem: unital ASH
algebras which are simple, have slow dimension growth, and in which projections
separate traces are determined up to isomorphism by their graded ordered
K-theory, and none of the latter three conditions can be relaxed in general.
