On surjectively universal Polish groups

TL;DR

This paper proves the existence of surjectively universal Polish groups using new metrics on free groups, providing multiple examples and conditions for their computation, thus answering a question posed by Kechris.

Contribution

It introduces new metrics on free groups to establish the existence of surjectively universal Polish groups and offers several examples and computational conditions.

Findings

Existence of surjectively universal Polish groups proven

Multiple examples of such groups provided

A sufficient condition for metric computability established

Abstract

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 \cite{DG}, 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.