The nuclear dimension of O∞-stable-C*-algebras

We show that every nuclear O∞-stable ∗ -homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, O∞-stable C ∗ -algebras have nuclear dimension 1. We also characterise when O∞-stable C ∗ -algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full O2-stable ∗ -homomorphisms have nuclear dimension 0