Published online by Cambridge University Press: 07 June 2023
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
$E({\mathbb {T}}^n)$.