Submission date: 29 February 2016.

Abstract:

We study, in an abstract context, equivalence relations which are invariant under group actions. More precisely, we fix a transformation group, and we study the orbital equivalence relations (i.e. orbit equivalence relations of normal subgroups) and a wider class of weakly orbital equivalence relations. For these sorts of relations we show (under some additional assumptions) that if each class is well-behaved, then so is the class space and the relation as a whole (where well-behaved means type-definable or closed for the classes and the relation and Hausdorff for the class space). We apply these conclusions in model theory to generalise a recent result tying type-definability and smoothness of invariant equivalence relations. We also obtain analogous results for type-definable actions (in model theory), as well as continuous actions of compact groups, or, more generally, proper group actions.

Mathematics Subject Classification: 54H15, 03C45, 22C05, 03E15

Keywords and phrases: transformation groups, equivalence relations, compact groups, bounded invariant equivalence relations, Borel cardinality

Full text arXiv 1602.09009: pdf, ps.