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Squares of Menger-bounded groups


Authors: Michał Machura, Saharon Shelah and Boaz Tsaban
Journal: Trans. Amer. Math. Soc. 362 (2010), 1751-1764
MSC (2000): Primary 54H11, 54C65, 03E17
DOI: https://doi.org/10.1090/S0002-9947-09-05169-1
Published electronically: November 16, 2009
MathSciNet review: 2574876
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Abstract: Using a portion of the Continuum Hypothesis, we prove that there is a Menger-bounded (also called $ o$-bounded) subgroup of the Baer-Specker group $ \mathbb{Z}^{\mathbb{N}}$, whose square is not Menger-bounded. This settles a major open problem concerning boundedness notions for groups and implies that Menger-bounded groups need not be Scheepers-bounded. This also answers some questions of Banakh, Nickolas, and Sanchis.


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Additional Information

Michał Machura
Affiliation: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland – and – Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel
Email: machura@ux2.math.us.edu.pl

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, Piscataway, New Jersey 08854
Email: shelah@math.huji.ac.il

Boaz Tsaban
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel – and – Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: tsaban@math.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-09-05169-1
Received by editor(s): May 1, 2007
Published electronically: November 16, 2009
Additional Notes: The authors were partially supported by the EU Research and Training Network HPRN-CT-2002-00287, United States-Israel BSF Grant 2002323, and the Koshland Center for Basic Research, respectively
Article copyright: © Copyright 2009 American Mathematical Society

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