Logic Seminar

For every Polish permutation group P≤Sym(N), let A↦[A]P be the assignment which maps every A⊆N to the set of all k∈N whose orbit under the action of the stabilizer PF of some finite F⊆A is finite. Then A↦[A]P is a closure operator and hence it endows P with a natural notion of dimension dim(P). This notion of dimension has been extensively studied in model theory when A↦[A]P satisfies additionally the exchange principle, that is, when A↦[A]P forms a pregeometry. However, under the exchange principle, every Polish permutation group P with dim(P)<∞ is locally compact and therefore unable to generate any "wild" dynamics. In this talk, we will discuss the relationship between dim(P) and certain strong ergodicity phenomena in the absence of the exchange principle. In particular, for every n∈N, we will provide a Polish permutation group P with dim(P)=n whose Bernoulli shift P↷RN is generically ergodic relative to the injective part of the Bernoulli shift of any permutation group Q with dim(Q)<n. We will use this to exhibit an equivalence relation of pinned cardinal ℵ1 which strongly resembles Zapletal's counterexample to a question of Kechris, but which does not Borel reduce to the latter. Our proofs rely on the theory of symmetric models of choiceless set theory and in the process we establish that a vast collection of symmetric models admit a theory of supports similar to the basic Cohen model. This is joint work with Assaf Shani.