On surjectively universal Polish groups

A Polish group is surjectively universal if it can be continuously homomorphically mapped onto every Polish group. Making use of a type of new metrics on free groups by Ding and Gao (2007) [3], we prove the existence of surjectively universal Polish groups, answering in the positive a question of Kechris. In fact, we give several examples of surjectively universal Polish groups.We find a sufficient condition to guarantee that the new metrics on free groups can be computed directly. We also compare this condition with CLI groups.