Basis Problem for Turbulent Actions II: c 0 -Equalities

Abstract

Let (Xn,dn) be a sequence of finite metric spaces of uniformly bounded diameter. An equivalence relation D on the product ∏ n X n defined by x → D y → if and only if lim sup n d n ( x n , y n ) = 0 is a c0-equality. A systematic study is made of c0-equalities and Borel reductions between them. Necessary and sufficient conditions, expressed in terms of combinatorial properties of metrics dn, are obtained for a c0-equality to be effectively reducible to the isomorphism relation of countable structures. It is proved that every Borel equivalence relation E reducible to a c0-equality D either reduces a c0-equality D' additively reducible to D, or is Borel-reducible to the equality relation on countable sets of reals. An appropriately defined sequence of metrics provides a c0-equality which is a turbulent orbit equivalence relation with no minimum turbulent equivalence relation reducible to it. This answers a question of Hjorth. It is also shown that, whenever E is an Fσ-equivalence relation and D is a c0-equality, every Borel equivalence relation reducible to both D and to E has to be essentially countable. 2000 Mathematics Subject Classification: 03E15.