A generalization of the ????0 dichotomy and a strengthening of the ????0ℕ dichotomy

Journal of Mathematical Logic 22 (1) (2022)
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Abstract

We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation.

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Benjamin Miller
University of Illinois, Urbana-Champaign

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