The small index property for homogeneous models in AECs

TL;DR

This paper extends the small index property theorem to strong amalgamation classes in AECs, including quasiminimal pregeometry structures, broadening its applicability beyond classical first-order theories.

Contribution

It generalizes the small index property to non-elementary classes, especially in the context of strong amalgamation and quasiminimal pregeometry structures.

Findings

Proves a small index property theorem for strong amalgamation classes.

Establishes the property for quasiminimal pregeometry structures.

Builds on and generalizes earlier results by Lascar and Shelah.

Abstract

We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah (in their case, for saturated models of uncountable first-order theories). We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.