Equivalence relations induced by actions of Polish groups
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- by Sławomir Solecki
- Trans. Amer. Math. Soc. 347 (1995), 4765-4777
- DOI: https://doi.org/10.1090/S0002-9947-1995-1311918-2
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Abstract:
We give an algebraic characterization of those sequences $({H_n})$ of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of ${H_0} \times {H_1} \times {H_2} \times \cdots$ are Borel. In particular, the equivalence relations induced by Borel actions of ${H^\omega }$, $H$ countable abelian, are Borel iff $H \simeq { \oplus _p}({F_p} \times \mathbb {Z}{({p^\infty })^{{n_p}}})$, where ${F_p}$ is a finite $p$-group, $\mathbb {Z}({p^\infty })$ is the quasicyclic $p$-group, ${n_p} \in \omega$, and $p$ varies over the set of all primes. This answers a question of R. L. Sami by showing that there are Borel actions of Polish abelian groups inducing non-Borel equivalence relations. The theorem also shows that there exist non-locally compact abelian Polish groups all of whose Borel actions induce only Borel equivalence relations. In the process of proving the theorem we generalize a result of Makkai on the existence of group trees of arbitrary height.References
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Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 4765-4777
- MSC: Primary 03E15; Secondary 04A15, 22A05
- DOI: https://doi.org/10.1090/S0002-9947-1995-1311918-2
- MathSciNet review: 1311918