On equivalence relations induced by locally compact abelian Polish groups

Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^{\omega }\mathrm{/}c(G)$, 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 $E(G){\le }_{B}E(H)$, then there is a continuous homomorphism $S:G_0\to H_0$ 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 $P(\omega )\mathrm{/}Fin$ can be embedded into Borel equivalence relations between $E({\mathbb{R}}^n)$ and $E({\mathbb{T}}^n)$.