Around the Small Index Property on quasiminimal classes

In the study of the connection between automorphism groups of models and the models themselves (or their theories, or their bi-interpretability class), the Small Index Property (SIP) has played a central role. The work of Hodges, Lascar, Shelah and Rubin among others has established in many cases when a model of a first order theory T has the Small Index Property.

With Ghadernezhad, we have studied this property for more general homogeneous classes. We have isolated properties of closure notions that allow to prove the SIP for some non-elementary cases, including Zilber’s pseudo-exponentiation and other examples.

I will present a panorama of these results, including our more recent generalizations of the Lascar-Shelah proof of SIP for uncountable structures. This last part is joint work with Zaniar Ghadernezhad.