The structure of random automorphisms of the random graph

TL;DR

This paper characterizes the size of conjugacy classes in the automorphism group of the random graph, revealing many non-Haar null classes and providing new insights into their structure using advanced measure-theoretic methods.

Contribution

It offers a complete description of conjugacy class sizes in the automorphism group of the random graph, including the existence of continuum many non-Haar null classes and a new proof of a classical result.

Findings

Many conjugacy classes are non-Haar null.

Every non-Haar null class contains a translated portion of every compact set.

Provides a new proof of Truss's classical result.

Abstract

We give a complete description of the size of the conjugacy classes of the automorphism group of the random graph with respect to Christensen's Haar null ideal. It is shown that every non-Haar null class contains a translated copy of a nonempty portion of every compact set and that there are continuum many non-Haar null conjugacy classes. Our methods also yield a new proof of an old result of Truss.