On the type-definability of the binding group in simple theories

Abstract

Let T be simple, work in 𝒞^eq over a boundedly closed set. Let p∈ S(∅) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose (p,Q) is ACL-embedded in Q (see definitions below). Then Aut(p/Q) with its action on p^𝒞 is type-definable in 𝒞^eq over ∅. In particular, if p∈ S(∅) is internal in a stably-embedded type-definable set Q, and p^𝒞∪ Q is stably-embedded, then Aut(p/Q) is type-definable with its action on p^𝒞.