Metrically universal abelian groups
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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
- MR Author ID: 984180
- Email: doucha@math.cas.cz
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5981-5998
- MSC (2010): Primary 22A05, 54H11, 03C98
- DOI: https://doi.org/10.1090/tran/7059
- MathSciNet review: 3646786