Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [BK] H. Becker and A. S. Kechris, Borel actions of Polish groups, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 334-341. MR 1185149 (93m:03083)
  • [F] L. Fuchs, Infinite abelian groups, vols. I and II, Academic Press, New York and London, 1970, 1973. MR 0255673 (41:333)
  • [L] D. Lascar, Why some people are excited by Vaught's conjecture, J. Symbolic Logic 50 (1985), 973-981. MR 820126 (87e:03089)
  • [M] M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 (1981), 301-318. MR 613284 (82m:03049)
  • [S] R. L. Sami, Polish group actions and the Vaught conjecture, Trans. Amer. Math. Soc. 341 (1994), 335-353. MR 1022169 (94c:03068)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03E15, 04A15, 22A05

Retrieve articles in all journals with MSC: 03E15, 04A15, 22A05

Additional Information

Keywords: Actions of Polish groups, equivalence relation induced by an action
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society