Non-Archimedean Polish groups involve S_∞ exactly when they satisfy a weakened disjoint amalgamation property, a weakened absolute generating indiscernibles property, or lack ordinal rank in a coanalytic rank function, and only such groups classify the benchmark equivalence relation =+.
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1 Pith paper cite this work. Polarity classification is still indexing.
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Classification Strength of Polish Groups: Involving $S_\infty$
Non-Archimedean Polish groups involve S_∞ exactly when they satisfy a weakened disjoint amalgamation property, a weakened absolute generating indiscernibles property, or lack ordinal rank in a coanalytic rank function, and only such groups classify the benchmark equivalence relation =+.