### 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_{∞} 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.

Original language | English |
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Pages (from-to) | 8829-8848 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 12 |

DOIs | |

Publication status | Published - Jan 1 2019 |

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### Keywords

- Amalgamation
- Automorphism group
- Christensen
- Compact catcher
- Conjugacy class
- Haar null
- Nonlocally compact Polish group
- Prevalent
- Random automorphism
- Shy
- Truss
- Typical element

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*371*(12), 8829-8848. https://doi.org/10.1090/tran/7758