The fine structure and Borel complexity of orbits

§0. Introduction Recent work in the general theory of Polish groups has looked for inspiration from the study of countable models, where a major tool in analyzing isomorphism types is provided by the Scott analysis. One has many results that were first proved in the case of the “logic action” on countable models by an appeal to the Scott analysis, and only later generalized to general Polish group actions – usually in some ad hoc manner, and occasionally in a somewhat diluted form. On top of this there were even a small number of results known for closed subgroups of S∞ by use of the Scott analysis but open for more general Polish groups. I will go through three different parallel definitions, each of which has some moral right to be considered the Scott analysis, but only the second of which historically fills this role.