  The motivation of this article is to introduce a kind of orbit equivalence
relations which can well describe structures and properties of Polish groups
from the perspective of Borel reducibility. 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 $G$.
  Let $G$ be a Polish group. (1) $G$ is a discrete countable group containing
at least two elements iff $E(G)\sim_BE_0$; (2) if $G$ is TSI uncountable
non-archimedean, then $E(G)\sim_BE_0^\omega$; (3) $G$ is non-archimedean iff
$E(G)\le_B=^+$; (4) let $G$ be a non-CLI Polish group and $H$ a CLI Polish
group, then $E(G)\not\le_BE_H^Y$ for any Polish $H$-space $Y$; (5) if $H$ is a
non-archimedean Polish group but $G$ is not, then $E(G)\not\le_BE_H^Y$ for any
Polish $H$-space $Y$.
  The notion of $\alpha$-unbalanced Polish group for $\alpha<\omega_1$ is
introduced. Let $G,H$ be Polish groups, $0<\alpha<\omega_1$. If $G$ is
$\alpha$-unbalanced but $H$ is not, then $E(G)\not\le_B E(H)$.
  For TSI Polish groups, the existence of Borel reduction is transformed into
the existence of a well-behaved continuous mapping between topological groups.
As its applications, for any Lie group $G$, denote $G_0$ the connected
component of the identity element $1_G$. Let $G$ and $H$ be two separable TSI
Lie groups. If $E(G)\le_BE(H)$, then there exists a continuous locally
injection $S:G_0\to H_0$. Moreover, if $G_0,H_0$ are abelian, $S$ is a group
homomorphism. Particularly, for $c_0,e_0,c_1,e_1\in{\mathbb N}$, $E({\mathbb
R}^{c_0}\times{\mathbb T}^{e_0})\le_BE({\mathbb R}^{c_1}\times{\mathbb
T}^{e_1})$ iff $e_0\le e_1$ and $c_0+e_0\le c_1+e_1$.
