ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS

Abstract

Abstract 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. The connected component of the identity of a Polish group G is denoted by $G_0$ . Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$ , then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is non-archimedean. The converse is also true when G is connected and compact. For $n\in {\mathbb {N}}^+$ , the partially ordered set $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$ .