Automorphism Groups of Countable Stable Structures

TL;DR

The paper constructs a stable structure with an automorphism group topologically isomorphic to that of any given countable structure, demonstrating stability cannot be inferred from the automorphism group's topology.

Contribution

It introduces a method to associate a stable structure to any countable structure with an isomorphic automorphism group, highlighting limitations in detecting stability via automorphism groups.

Findings

Automorphism groups of countable structures can be topologically isomorphic regardless of stability.

Stability of a structure is not reflected in the topological properties of its automorphism group.

The construction applies to all countable structures, showing a fundamental limitation in model-theoretic classification.

Abstract

For every countable structure $M$ we construct an $\aleph_0$-stable countable structure $N$ such that $Aut(M)$ and $Aut(N)$ are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure $M$ from the topological properties of the Polish group $Aut(M)$.