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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The structure of random automorphisms of countable structures


Authors: Udayan B. Darji, Márton Elekes, Kende Kalina, Viktor Kiss and Zoltán Vidnyánszky
Journal: Trans. Amer. Math. Soc. 371 (2019), 8829-8848
MSC (2010): Primary 03E15, 22F50; Secondary 03C15, 28A05, 54H11, 28A99
DOI: https://doi.org/10.1090/tran/7758
Published electronically: February 25, 2019
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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.


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Additional Information

Udayan B. Darji
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292; and Ashoka University, Rajiv Gandhi Education City, Kundli, Rai 131029, India
Email: ubdarj01@louisville.edu

Márton Elekes
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary; and Eötvös Loránd University, Institute of Mathematics, Pázmány Péter sétány 1/c, 1117 Budapest, Hungary
Email: elekes.marton@renyi.mta.hu

Kende Kalina
Affiliation: Eötvös Loránd University, Institute of Mathematics, Pázmány Péter sétány 1/c, 1117 Budapest, Hungary
Email: kkalina@cs.elte.hu

Viktor Kiss
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary; and Eötvös Loránd University, Institute of Mathematics, Pázmány Péter sétány 1/c, 1117 Budapest, Hungary
Email: kiss.viktor@renyi.mta.hu

Zoltán Vidnyánszky
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090 Wien, Austria; and Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary
Email: zoltan.vidnyanszky@univie.ac.at

DOI: https://doi.org/10.1090/tran/7758
Keywords: Nonlocally compact Polish group, Haar null, Christensen, shy, prevalent, typical element, automorphism group, compact catcher, Truss, amalgamation, random automorphism, conjugacy class
Received by editor(s): August 24, 2018
Received by editor(s) in revised form: November 10, 2018
Published electronically: February 25, 2019
Additional Notes: The second, fourth, and fifth authors were partially supported by the National Research, Development and Innovation Office—NKFIH, grants no. 113047, no. 104178, and no. 124749.
The fifth author was also supported by FWF grant no. P29999.
Article copyright: © Copyright 2019 American Mathematical Society