Forcing and Equivalence Relations E(Λ≀Θ)

Given a Polish group G, let E(G) be the right coset equivalence relation G(omega)/c(G), where c(G) is the group of all convergent sequences in G. In this article, we use the tool of the forcing method to prove a rigid theorem for the wreath product Lambda(sic)Theta: Let Lambda, Gamma, Theta, Theta ' be four nontrivial countable discrete groups. Suppose Lambda has no finite subgroup. Then E(Lambda(sic)Theta) <=(B) E(Gamma (sic) Theta ') if and only if there is a group isomorphism phi : Lambda -> (Gamma) over tilde/Delta, where (Gamma) over tilde is a subgroup of Gamma, Delta is normal in (Gamma) over tilde. A direct corollary is that, there are continuum many non-archimedean Polish groups (G(r))(r is an element of R) such that these E(G(r))'s are pairwise Borel incomparable. Using the technique developed in this article, we also prove (1) E(Z (sic) Z(2))<(B) E(Z (sic) Z(2)(omega)). (2) E(Z (sic) Z(2))<(B) E(Z (sic) Z(2))(2)