Classification Strength of Polish Groups: Involving $S_\infty$

Abstract

In recent years, much work has been done to measure and compare the complexity of orbit equivalence relations, especially for certain classes of Polish groups. We start by introducing some language to organize this previous work, namely the notion of \textbf{classification strength} of Polish groups. Broadly speaking, a Polish group $G$ has stronger classification strength than $H$ if every orbit equivalence relation induced by a continuous action of $H$ on a Polish space can be ``emulated" by such an action of $G$ in the sense of Borel reduction. Among the non-Archimedean Polish groups, the groups with the highest classification strength are those that involve $S_\infty$, the Polish group of permutations of a countably-infinite set. We prove that several properties, including a weakening of the disjoint amalgamation in Fra\"{i}ss\'{e} theory, a weakening of the existence of an absolute set of generating indiscernibles, and not having ordinal rank in a particular coanalytic rank function, are all equivalent to a non-Archimedean Polish group involving $S_\infty$. Furthermore, we show the equivalence relation $=^+$, which is a relatively simple benchmark equivalence relation in the theory of Borel reducibility, can only be classified by such groups that involve $S_\infty$.