Complete orbit equivalence relation and non-universal Polish groups
Longyun Ding, Ruiwen Li, Bo Peng
TL;DR
The paper answers Sabok's question in the affirmative by showing that there exists a non-universal Polish group that induces a complete orbit equivalence relation. It builds on the construction of surjectively universal Polish groups via free groups with Graev-type metrics and scales, establishing a surjectively universal group $ar{F}_oldsymbol{ abla}( abla)$ that is not universal. By leveraging a continuous surjective homomorphism from a surjectively universal group to a Polish space with a group action, the authors obtain a complete orbit equivalence relation arising from such an action. The result demonstrates a precise link between orbit equivalence completeness and group universality, providing a positive answer to Sabok's problem and highlighting the role of surjectively universal groups in this context.
Abstract
We show that a non-universal Polish group can induce a complete orbit equivalence relation, which answers a question of Sabok from \cite{OPENPROBLEMS}.