 
Summary: Countable Borel Equivalence Relations III
Simon Thomas
Rutgers University
17th November 2007
Simon Thomas (Rutgers University) Appalachian Set Theory Workshop 17th November 2007
A quick recap
The Fundamental Question
Let G be a countable group and let X be a standard Borel Gspace.
To what extent does the data (X, EX
G ) "remember" the group G and
its action on X?
More accurately, to what extent does the data (C, EX
G C) "remember"
the group G and its action on X, where C is an arbitary Borel complete
section?
Further Hypotheses
We shall usually also assume that:
G acts freely on X; i.e. g · x = x for all 1 = g G and x X.
There exists a Ginvariant probability measure µ on X.
Simon Thomas (Rutgers University) Appalachian Set Theory Workshop 17th November 2007
