The Strong Small Index Property for Free Homogeneous Structures

TL;DR

This paper proves that certain countable homogeneous structures with a free stationary independence relation possess the strong small index property, expanding understanding of automorphism groups in model theory.

Contribution

It establishes that in algebraically locally finite structures, the small index property implies the strong version, and applies this to new classes of countable free homogeneous structures.

Findings

Countable free homogeneous structures in a locally finite language have the strong small index property.

New continuum-sized classes of -categorical structures with non-isomorphic automorphism groups.

The strong small index property holds under specific conditions in algebraically locally finite structures.

Abstract

We show that in algebraically locally finite countable homogeneous structures with a free stationary independence relation the small index property implies the strong small index property. We use this and the main result of [15] to deduce that countable free homogeneous structures in a locally finite relational language have the strong small index property. We also exhibit new continuum sized classes of $\aleph_0$-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.