Metrically universal abelian groups

Abstract

We give a positive answer to the question of Shkarin (\emph{On universal abelian topological groups}, 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.

The last version's only change is the author's grant information. The new version contains simplified proofs and other changes based on the referee's comments