A Borel parametrization of Polish groups (Q1084304)

This paper constructs a standard Borel space, PG, and a map \(p\in PG\to G(p)\) such that each G(p) is a Polish group, and such that every Polish group is isomorphic to at least one of the groups G(p); PG thus serves as a parameter space for all Polish groups. We formulate the notion of a Borel map from a standard B space to Polish groups, and that of a Borel functor from a standard Borel groupoid to Polish groups; both are defined in terms of the existence of Borel factorizations through PG. We apply these ideas to establish a general ''cohomology lemma'', asserting that cocycles, with values in Borel family of Polish groups, may be cobounded into a given family of dense, normal, Borel subgroups, whenever the underlying groupoid is a hyperfinite equivalence relation.