1
NEW BOREL INDEPENDENCE RESULTS
by
Harvey M. Friedman
April 30, 2007
May 7, 2007
sketch
1. Introduction.
2. Countable Borel equivalence relations on .
3. Countable Borel equivalence relations on
2
.
4. Degrees into finitely generated groups.
5. Borel functions on groups.
6. Countable Borel quasi orders.
1. INTRODUCTION.
A considerable amount of interdisciplinary work has been
done on countable Borel equivalence relations. See [JKL02].
Many researchers in logic have worked on countable Borel
equivalence relations, who have utilized the work of many
people in ergodic theory and related areas. These include
the following names taken from the list of references of
[JKL02]:
S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo,
C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes.
C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A.
Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P.
de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S.
Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore,
M.G. Nadkarni, C. Nebbia,
A.L.T. Patterson,
U. Krengel,
A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman,
Solecki, R. Spatzier, J. Steel, D. Sullivan, S. Thomas, A.
Valette, V.S. Varadarajan, B. Velickovic, B. Weiss, J.D.M.
Wright, R.J. Zimmer.
Many thanks to Simon Thomas for advice on this manuscript.
A Borel equivalence relation is a pair (X,E), where X is a
Polish space (complete separable metric space), and E is a
Borel equivalence relation on X.
A countable Borel equivalence relation is a Borel
equivalence relation in which every equivalence class is
countable.