Internality and interpretable automorphism groups in simple theories
The paper investigates some possible partial extensions of the binding group theorem from stable to simple theories. For \(T\) a simple theory, \({\mathcal Q}\) an \(\emptyset\)-invariant subset of the monster model \({\mathcal C}\) of \(T\) and \(p\) a \({\mathcal Q}\)-internal type over \(\emptyset\), the automorphism group \(G = \Aut(p({\mathcal C})/{\mathcal Q})\) is considered, a ``small'' normal subgroup \(G_0^+\) is introduced, and it is shown that \(G_0^+\) is of finite exponent and that the quotient group \(G/G_0^+\) is interpretable in an invariant way. Further results are proved under stronger assumptions on \({\mathcal Q}\) and \(p\). For instance, it is shown that, if \(p\) satisfies some suitable internality or multiplicity 1 conditions and \({\mathcal Q}\) is pseudo-open, then \(G\) is type-definable.
scientific article; zbMATH DE number 2117329