The structure of random automorphisms

Udayan B. Darji, Márton Elekes, Kende Kalina, Viktor Kiss, Zoltán Vidnyánszky

Published 2017-05-22Version 1

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 behaviour of the automorphisms is entirely different from the Baire category case. We generalise 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. A complete description of the non-Haar null conjugacy classes of the automorphism groups of $(\mathbb{Q},<)$ and of the random graph is given, in fact, we prove that every non-Haar null class contains a translated copy of a non-empty portion of every compact set. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.