Conjugacy class conditions in locally compact second countable groups

Wesolek, Phillip

doi:10.1090/proc/12645

Conjugacy class conditions in locally compact second countable groups
HTML articles powered by AMS MathViewer

by Phillip Wesolek

Proc. Amer. Math. Soc. 144 (2016), 399-409

DOI: https://doi.org/10.1090/proc/12645

Published electronically: August 18, 2015

PDF | Request permission

Abstract:

Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for $S_{\infty }$, $Aut(\mathbb {Q},<)$, and, less trivially, $Aut(\mathcal {R})$ for $\mathcal {R}$ the random graph. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.

References

Ethan Akin, Eli Glasner, and Benjamin Weiss, Generically there is but one self homeomorphism of the Cantor set, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3613–3630. MR 2386239, DOI 10.1090/S0002-9947-08-04450-4
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation. MR 1728312
Anton Deitmar and Siegfried Echterhoff, Principles of harmonic analysis, Universitext, Springer, New York, 2009. MR 2457798
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 32641, DOI 10.1112/jlms/s1-24.4.247
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Alexander S. Kechris and Christian Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350. MR 2308230, DOI 10.1112/plms/pdl007
Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 (2000), no. 1, 1–7. MR 1764885, DOI 10.1007/s000130050466
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132, DOI 10.1007/978-3-642-01642-4
J. K. Truss, Generic automorphisms of homogeneous structures, Proc. London Math. Soc. (3) 65 (1992), no. 1, 121–141. MR 1162490, DOI 10.1112/plms/s3-65.1.121
John S. Wilson, On the structure of compact torsion groups, Monatsh. Math. 96 (1983), no. 1, 57–66. MR 721596, DOI 10.1007/BF01298934
John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054
E. I. Zel′manov, On periodic compact groups, Israel J. Math. 77 (1992), no. 1-2, 83–95. MR 1194787, DOI 10.1007/BF02808012