The conjugacy problem for the automorphism group of the random graph

TL;DR

This paper proves that determining whether two automorphisms of the random graph are conjugate is a highly complex problem, classified as Borel complete, and explores similar issues in other countably categorical structures.

Contribution

It establishes the Borel completeness of the conjugacy problem for the automorphism group of the random graph and discusses analogous problems for other structures.

Findings

Conjugacy problem for the automorphism group of the random graph is Borel complete.

Discusses the conjugacy problem for other countably categorical structures.

Highlights the complexity of automorphism conjugacy in model theory.

Abstract

We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.