Actions by the infinite permutation group on N
by
Greg Hjorth
UCLA

A space is Polish if it is seperable and its topology admits a complete metric. A topological group is a Polish group if it is Polish as a space. Examples here include: l², Hilbert space; S_\infty, the (countably) infinite permutation group; R^N with the product structure. If G is a Polish group acting continuously on a Polish space X, then we say that X is a Polish S_\infty-space and use E^X_G to denote the orbit equivalence relation - x E^X_G y iff there exists a g in G (g ·x = y).

For E and F equivalence relations on Polish spaces X and Y, we say that E is Borel reducible to F, written E <= _B F, if there is a Borel function \theta: X --> Y such that for every x, y in X (xEy iff \theta(x) F \theta(y)). For instance, Becker and Kechris have shown that if L is the language generated by a single binary relation, and if Mod(L) is the space of all L-structures with underlying set N, then for any Polish S_\infty-space X we have E^X_G <= _B =~ |_Mod(L).

I will discuss some results that try to understand for which Polish groups we always have that all the orbit equivalence relations are Borel reducible to one induced by S_\infty, and are hence Borel reducible to =~ |_Mod(L) by the Becker-Kechris result above. It turns out that with groups such as R^N such a reduction is always possible, but with groups such as l² it is in general not.

Date received: March 18, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caac-13.