Conjugacy for homogeneous ordered graphs

TL;DR

This paper proves that the conjugacy problem for automorphisms of any countable homogeneous ordered graph is as complex as the most complicated classification problems, by establishing a strong extension property called ABAP.

Contribution

It introduces the ABAP extension property for homogeneous ordered graphs and shows it leads to Borel completeness of the conjugacy problem.

Findings

Conjugacy problem for automorphisms is Borel complete.

Homogeneous ordered graphs satisfy the ABAP extension property.

Isomorphism on substructures reduces to conjugacy on automorphisms.

Abstract

We show that for any countable homogeneous ordered graph $G$, the conjugacy problem for automorphisms of $G$ is Borel complete. In fact we establish that each such $G$ satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of $G$ is Borel reducible to the conjugacy relation on automorphisms of $G$.