The structure of random automorphisms of countable structures

TL;DR

This paper explores the measure-theoretic size of conjugacy classes in automorphism groups of countable structures, revealing that typical automorphisms behave differently under Haar null measure than in the Baire category sense.

Contribution

It generalizes Dougherty and Mycielski's theorems from $S_$ to broader automorphism groups, introducing the Cofinal Strong Amalgamation Property and analyzing measure-theoretic properties.

Findings

Automorphism groups can be decomposed into a union of meager and Haar null sets.

The behavior of typical automorphisms differs significantly under Haar null measure.

The results extend measure-theoretic understanding of automorphism groups beyond the Baire category framework.

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. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about $S_\infty$ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the Cofinal Strong Amalgamation Property. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.