Projectively universal countable metrizable groups (Q2817026)

A topological group is \textit{Polish} if it is homeomorphic to a complete separable metric space. As shown in [the second author, Funct. Anal. Appl. 20, 160--161 (1986); translation from Funkts. Anal. Prilozh. 20, No. 2, 86--87 (1986; Zbl 0608.22003)]; the second author, Comment. Math. Univ. Carolin. 31, 181--182 (1990; Zbl 0699.54011); \textit{Itaï Ben Yaacov}, Proc. Am. Math. Soc. 142, 2459--2467 (2014; Zbl 1311.46004)], there exist \textit{injectively universal Polish groups}, that is, Polish groups \(G\) such that every Polish group \(H\) is topological isomorphic to a closed subgroup of \(G\). Recently, in [\textit{Longyun Ding}, Adv. Math. 231, 2557--2572 (2012; Zbl 1257.03076)] an example is given of a \textit{couniversal Polish group} (or \textit{projectively universal Polish group}), that is, a Polish group \(G\) such that every Polish group \(H\) is topological isomorphic to a topological quotient group of \(G\).NEWLINENEWLINEIn the paper under review, two constructions are provided of \textit{couniversal countable metrizable groups}, that is, countable metrizable groups \(G\) such that every countable metrizable group \(H\) is topological isomorphic to a topological quotient group of \(G\). Moreover, the Rajkov completion of a couniversal countable metrizable group is shown to be a couniversal Polish group.