The possibility of designing efficient, ex-post incentive compatible, single valued direct mechanisms crucially depends on the domain of types and preferences on which they are defined. In a general framework that allows for interdependent types, we identify two relevant classes of domains. For the class of those that we call knit, we show that only the constant mechanisms can be ex post (or even interim) incentive compatible. We then propose a concept of ex post group incentive compatibility that implies Pareto efficiency on the range of mechanisms, and show that this strong condition will be implied by standard ex post incentive compatibility in our second class of domains, that we call partially knit. That opens the door to identify environments in which good incentives are not at odds with efficiency. In the particular case of private values, our results provide conditions under which individual and strong group strategy-proofness become equivalent. We provide examples of voting, matching, and auction mechanisms to which our theorems apply.