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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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