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The structure of random automorphisms of the rational numbers

Volume 250 / 2020

Udayan B. Darji, Márton Elekes, Kende Kalina, Viktor Kiss, Zoltán Vidnyánszky Fundamenta Mathematicae 250 (2020), 1-20 MSC: Primary 03E15, 22F50; Secondary 03C15, 28A05, 54H11, 28A99. DOI: 10.4064/fm618-9-2019 Published online: 21 February 2020

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 $\operatorname{Aut} (\mathbb Q , \lt )$, 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 the group $\operatorname{Aut} (\mathbb Q , \lt )$ 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.

Authors

  • Udayan B. DarjiDepartment of Mathematics
    University of Louisville
    Louisville, KY 40292, U.S.A.
    and
    Ashoka University
    Rajiv Gandhi Education City
    Kundli, Rai 131029, India
    http://www.math.louisville.edu/~darji
    e-mail
  • Márton ElekesAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    PO Box 127
    1364 Budapest, Hungary
    and
    Institute of Mathematics
    Eötvös Loránd University
    Pázmány Péter s. 1/c
    1117 Budapest, Hungary
    http://www.renyi.hu/~emarci
    e-mail
  • Kende KalinaInstitute of Mathematics
    Eötvös Loránd University
    Pázmány Péter s. 1/c
    1117 Budapest, Hungary
    e-mail
  • Viktor KissAlfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    PO Box 127
    1364 Budapest, Hungary
    and
    Institute of Mathematics
    Eötvös Loránd University
    Pázmány Péter s. 1/c
    1117 Budapest, Hungary
    e-mail
  • Zoltán VidnyánszkyKurt Gödel Research Center for Mathematical Logic
    Universität Wien
    Währinger Strasse 25
    1090 Wien, Austria
    and
    Alfréd Rényi Institute of Mathematics
    Hungarian Academy of Sciences
    PO Box 127
    1364 Budapest, Hungary
    http://www.logic.univie.ac.at/~vidnyanszz77
    e-mail

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