Reducibility and nonreducibility between

equivalence relations

Authors:
Randall Dougherty and Greg Hjorth

Journal:
Trans. Amer. Math. Soc. **351** (1999), 1835-1844

MSC (1991):
Primary 04A15, 03E15; Secondary 46B45

Published electronically:
January 26, 1999

MathSciNet review:
1671377

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that, for , the relation of -equivalence between infinite sequences of real numbers is Borel reducible to the relation of -equivalence (i.e., the Borel cardinality of the quotient is no larger than that of ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.

**1.**Howard Becker and Alexander S. Kechris,*The descriptive set theory of Polish group actions*, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR**1425877****2.**K. J. Falconer and D. T. Marsh,*Classification of quasi-circles by Hausdorff dimension*, Nonlinearity**2**(1989), no. 3, 489–493. MR**1005062****3.**L. A. Harrington, A. S. Kechris, and A. Louveau,*A Glimm-Effros dichotomy for Borel equivalence relations*, J. Amer. Math. Soc.**3**(1990), no. 4, 903–928. MR**1057041**, 10.1090/S0894-0347-1990-1057041-5**4.**G. Hjorth,*Actions by the classical Banach spaces*, preprint, UCLA, 1996.**5.**Alexander S. Kechris,*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597****6.**H. von Koch,*On a continuous curve without tangents constructible from elementary geometry*, Classics on fractals (G. Edgar, ed.), Addison-Wesley, Reading, Massachusetts, 1993, pp. 25-45.**7.**Alain Louveau and Boban Veličković,*A note on Borel equivalence relations*, Proc. Amer. Math. Soc.**120**(1994), no. 1, 255–259. MR**1169042**, 10.1090/S0002-9939-1994-1169042-2**8.**Claude Tricot,*Curves and fractal dimension*, Springer-Verlag, New York, 1995. With a foreword by Michel Mendès France; Translated from the 1993 French original. MR**1302173**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
04A15,
03E15,
46B45

Retrieve articles in all journals with MSC (1991): 04A15, 03E15, 46B45

Additional Information

**Randall Dougherty**

Affiliation:
Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Email:
rld@math.ohio-state.edu

**Greg Hjorth**

Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095-1555

Email:
greg@math.ucla.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02261-8

Keywords:
Borel equivalence relations,
reducibility,
Borel cardinality

Received by editor(s):
April 4, 1997

Received by editor(s) in revised form:
May 11, 1997

Published electronically:
January 26, 1999

Additional Notes:
The first author was partially supported by NSF grant number DMS-9158092. The second author was partially supported by NSF grant number DMS-9622977.

Article copyright:
© Copyright 1999
American Mathematical Society