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)

 
 

 

An example of a non non-archimedean Polish group with ample generics


Author: Maciej Malicki
Journal: Proc. Amer. Math. Soc. 144 (2016), 3579-3581
MSC (2010): Primary 03E15, 54H11
DOI: https://doi.org/10.1090/proc/13017
Published electronically: March 30, 2016
MathSciNet review: 3503726
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For an analytic $ P$-ideal $ I$ we denote by $ S_I$ the Polish group of all the permutations of $ \mathbbm {N}$ with support in $ I$, equipped with a topology given by the corresponding submeasure on $ I$. We show that if $ \mbox {Fin} \subsetneq I$, then $ S_I$ has ample generics. This implies that there exists a non non-archimedean Polish group with ample generics.


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

  • [1] Wilfrid Hodges, Ian Hodkinson, Daniel Lascar, and Saharon Shelah, The small index property for $ \omega $-stable $ \omega $-categorical structures and for the random graph, J. London Math. Soc. (2) 48 (1993), no. 2, 204-218. MR 1231710 (94d:03063), https://doi.org/10.1112/jlms/s2-48.2.204
  • [2] A. Kaïchouh and F. Le Maître, Connected Polish groups with ample generics, arXiv:1503.04298.
  • [3] A. Kechris, Dynamics of non-archimedean Polish groups, European Congress of Mathematics: Krakow, July 2-7, 2012 (ed. R. Latala and A. Rucinski; European Mathematical Society, 2013) 375-397.
  • [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 (2008a:03079), https://doi.org/10.1112/plms/pdl007
  • [5] Sławomir Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), no. 1-3, 51-72. MR 1708146 (2000g:03112), https://doi.org/10.1016/S0168-0072(98)00051-7
  • [6] Sławomir Solecki, Local inverses of Borel homomorphisms and analytic P-ideals, Abstr. Appl. Anal. 3 (2005), 207-219. MR 2197115 (2006k:22002), https://doi.org/10.1155/AAA.2005.207
  • [7] Todor Tsankov, Compactifications of $ \mathbb{N}$ and Polishable subgroups of $ S_\infty $, Fund. Math. 189 (2006), no. 3, 269-284. MR 2213623 (2007c:54022), https://doi.org/10.4064/fm189-3-4

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E15, 54H11

Retrieve articles in all journals with MSC (2010): 03E15, 54H11


Additional Information

Maciej Malicki
Affiliation: Department of Mathematics and Mathematical Economics, Warsaw School of Economics, al. Niepodleglosci 162, 02-554,Warsaw, Poland
Email: mamalicki@gmail.com

DOI: https://doi.org/10.1090/proc/13017
Keywords: Ample generics, non-archimedean groups, $P$-ideals
Received by editor(s): March 17, 2015
Received by editor(s) in revised form: October 23, 2015
Published electronically: March 30, 2016
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society