Universal and ultrahomogeneous abelian Polish metric group

Abstract

We use Fra\" iss\' e theoretic methods to construct a universal and ultrahomogeneous abelian separable metric group. We show that such a group is a universal abelian Polish group, thus we provide another proof of a result already discovered by Shkarin. We then construct universal and ultrahomogeneous abelian periodic metric groups with bounded exponents; it follows that these groups are also universal objects for the corresponding classes of abelian Polish groups.