Computable Polish Group Actions

Using methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. Among other results, we provide a new recursion-theoretic characterization of Σ3-orbits in Gspaces, and we also give a sufficient condition for an orbit under effective G-action to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.