Projectively universal countable metrizable groups
  We prove that there exists a countable metrizable topological group $G$ such that every countable metrizable group is isomorphic to a quotient of $G$. The completion $H$ of $G$ is a Polish group such that every Polish group is isomorphic to a quotient of $H$. 
Vladimir G. Pestov, Vladimir V. Uspenskij
math.GR