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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Equivalence relations induced by actions of Polish groups


Author: Sławomir Solecki
Journal: Trans. Amer. Math. Soc. 347 (1995), 4765-4777
MSC: Primary 03E15; Secondary 04A15, 22A05
MathSciNet review: 1311918
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1311918-2
PII: S 0002-9947(1995)1311918-2
Keywords: Actions of Polish groups, equivalence relation induced by an action
Article copyright: © Copyright 1995 American Mathematical Society