The structure of random automorphisms of the rational numbers

TL;DR

This paper explores the measure-theoretic size of conjugacy classes in the automorphism group of the rational numbers, revealing a rich structure with many non-Haar null classes, contrasting with the Baire category perspective.

Contribution

It provides a complete measure-theoretic classification of conjugacy classes in Aut(Q, <), highlighting the existence of continuum many non-Haar null classes, which differs from the Baire category approach.

Findings

Existence of continuum many non-Haar null conjugacy classes.

Contrast between measure-theoretic and Baire category notions of typicality.

Complete description of conjugacy class sizes in Aut(Q, <).

Abstract

In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particular, we show that there exist continuum many non-Haar null conjugacy classes, illustrating that the random behaviour is quite different from the typical one in the sense of Baire category.