Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Conjugacy class conditions in locally compact second countable groups

Author: Phillip Wesolek
Journal: Proc. Amer. Math. Soc. 144 (2016), 399-409
MSC (2010): Primary 22D05, 03E15
Published electronically: August 18, 2015
MathSciNet review: 3415606
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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 (2008m:22009),
  • [2] N. 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 (2001g:17006)
  • [3] Anton Deitmar and Siegfried Echterhoff, Principles of harmonic analysis, Universitext, Springer, New York, 2009. MR 2457798 (2010g:43001)
  • [4] 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 (81k:43001)
  • [5] Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247-254. MR 0032641 (11,322d)
  • [6] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • [7] 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 (2008a:03079),
  • [8] Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556 (2003e:00003)
  • [9] L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 (2000), no. 1, 1-7. MR 1764885 (2001i:20059),
  • [10] Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104 (17,383b)
  • [11] 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 (2011a:20058)
  • [12] J. K. Truss, Generic automorphisms of homogeneous structures, Proc. London Math. Soc. (3) 65 (1992), no. 1, 121-141. MR 1162490 (93f:20008),
  • [13] John S. Wilson, On the structure of compact torsion groups, Monatsh. Math. 96 (1983), no. 1, 57-66. MR 721596 (85a:22007),
  • [14] John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054 (2000j:20048)
  • [15] E. I. Zelmanov, On periodic compact groups, Israel J. Math. 77 (1992), no. 1-2, 83-95. MR 1194787 (94e:20055),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 22D05, 03E15

Retrieve articles in all journals with MSC (2010): 22D05, 03E15

Additional Information

Phillip Wesolek
Affiliation: Université Catholique de Louvain, Institut de Recherche en Mathématiques et Physique (IRMP), Chemin du Cyclotron 2, box L7.01.02, 1348 Louvain-la-Neuve, Belgique

Keywords: Totally disconnected locally compact groups, profinite groups, comeagre conjugacy class
Received by editor(s): November 25, 2013
Received by editor(s) in revised form: September 17, 2014
Published electronically: August 18, 2015
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society