Logic Seminar

Full groups were introduced in Dye's visionary paper of 1959 as subgroups of Aut(X,µ) stable under cutting and pasting their elements along a countable partition of the probability space (X,µ). Two well-known examples are Aut(X,µ) itself and full groups of countable measure preserving equivalence relations, whose Borel complexity  (as subsets of Aut(X,µ)) was determined by Wei in 2005.

We will discuss other full groups, see that a lot of them are Borel subgroups of Aut(X,µ), and find a compatible Polish group topology on orbit full groups, which are full groups of equivalence relations induced by Borel actions of Polish groups. This talk is based on a joint work with Carderi.