Actions of non-compact and non-locally compact Polish groups (Q2710616)

The orbit equivalence relation~\(E^X_G\) of a~group~\(G\) acting on a~space~\(X\) is defined by \(xE^X_Gy\) if and only if \(y=g\cdot x\) for some \(g\in G\). The author proves that if \(G\)~is a~non-compact Polish group and \(X\)~is a~Polish space then there is a~continuous free action of~\(G\) on~\(X\) with non-smooth orbit equivalence relations, i.e., \(E^X_G\)~is not Borel reducible to the equality relation on a~Polish space. This answers affirmatively a~question of Kechris. The author establishes results related to local compactness of the group with a~continuous action on a~Polish space whose orbit equivalence relation is not reducible to a~countable Borel equivalence relation. Generalizing a~result of Hjorth he proves that each infinite dimensional separable Banach space has a~continuous action on a~Polish space with non-Borel orbit equivalence relation. This provides a~characterization of local compactness in separable Banach spaces.