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Metrically universal abelian groups


Author: Michal Doucha
Journal: Trans. Amer. Math. Soc. 369 (2017), 5981-5998
MSC (2010): Primary 22A05, 54H11, 03C98
DOI: https://doi.org/10.1090/tran/7059
Published electronically: April 24, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a positive answer to the question of Shkarin (Mat. Sb. 190 (1999), no. 7, 127-144) whether there exists a metrically universal abelian separable group equipped with invariant metric.

Our construction also gives an example of a group structure on the Urysohn universal space that is substantially different from the previously known examples. Under some cardinal arithmetic assumptions, our results generalize to higher cardinalities.


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Additional Information

Michal Doucha
Affiliation: Institute of Mathematics, Academy of Sciences, Prague, Czech Republic – and – Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Email: doucha@math.cas.cz

DOI: https://doi.org/10.1090/tran/7059
Keywords: Universal group, Fra\"{\i}ss\'e theory, Urysohn space
Received by editor(s): January 15, 2014
Received by editor(s) in revised form: January 2, 2015, March 16, 2015, June 1, 2015, December 15, 2015, August 9, 2016, and September 6, 2016
Published electronically: April 24, 2017
Additional Notes: The research of the author was partially supported by the grant IAA100190902 of the Grant Agency of the Academy of Sciences of the Czech Republic and by IMPAN’s international fellowship programme partially sponsored by PCOFUND-GA-2012-600415.
Article copyright: © Copyright 2017 American Mathematical Society

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