The conjugacy problem for automorphism groups of homogeneous digraphs

TL;DR

This paper determines the complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs, establishing a dichotomy that classifies the problem as either very simple or very complex.

Contribution

It completes the classification of the Borel complexity of conjugacy problems for automorphism groups of homogeneous structures, extending previous results to digraphs.

Findings

The conjugacy problem complexity is either minimal or maximal among classifiable relations.

A dichotomy theorem is established for the complexity of these problems.

Discussion on extending results to broader classes of homogeneous structures.

Abstract

We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.