Interpretable Groups, Stably Embedded Sets, and Vaughtian Pairs

The paper concerns sufficiently saturated structures M over a countable language with a unary predicate P . It is shown that if P ( M )is stably embedded and there are no Vaughtian pairs with respect to P , then an infinite group is interpretable over M (in an infinitary sense of ‘interpretable’). Also, it is shown that if M is ω‐categorical, f : D → P is a 0‐definable map with finite fibres, and P ( M ) is stably embedded but D is not, then some infinite group is first‐order interpretable over M .