The structure of random automorphisms of the rational numbers
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
