Groups of quasi-invariance and the Pontryagin duality

A Polish group $G$ is called a group of quasi-invariance or a QI-group, if there exist a locally compact group $X$ and a probability measure $\mu$ on $X$ such that 1) there exists a continuous monomorphism of $G$ to $X$, and 2) for each $g\in X$ either $g\in G$ and the shift $\mu_g$ is equivalent to $\mu$ or $g\not\in G$ and $\mu_g$ is orthogonal to $\mu$. It is proved that $G$ is a $\sigma$-compact subset of $X$. We show that there exists a quotient group $\mathbb{T}^H_2$ of $\ell^2$ modulo a discrete subgroup which is a Polish monothetic non locally quasi-convex (and hence nonreflexive) pathwise connected QI-group, and such that the bidual of $\mathbb{T}^H_2$ is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of $X$.