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 $E({\mathbb {T}}^n)$. (shrink)

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. We first established two results: (1) Let $G,H$ be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$. (2) Let G be a Polish group. Then the following are equivalent: (a) G is TSI non-archimedean; (b) $E(G)\leq _B E_0^\omega $ ; and (c) $E(G)\leq _B {\mathbb {R}}^\omega /c_0$. In (...) particular, $E(G)\sim _B E_0^\omega $ iff G is TSI uncountable non-archimedean. A critical theorem presented in this article is as follows: Let G be a TSI Polish group, and let H be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le _BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker (S)$ is non-archimedean, where $G_0$ is the connected component of the identity of G. The converse holds if G is connected, $S(G)$ is closed in H, and the interval $[0,1]$ can be embedded into H. As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively. (shrink)