Model theory of divisible rigid groups

A solvable group $G$ is called rigid, more precisely $m$-rigid, if there exists a normal series of subgroups
$$G\ =\ G_1\ >\ G_2\ >\ \ldots\ >\ G_m\ >\ G_{m+1}\ =\ 1 ,$$
where all quotients $G_i/G_{i+1}$ are abelian and when viewed as right modules over $\mathbb{Z} \left[ G/G_i \right]$, do not have torsion. Free solvable groups and iterated wreath products of torsion-free abelian groups are rigid, as well as their subgroups. A rigid group $G$ is termed divisible if elements of the quotient $G_i/G_{i+1}$ are divisible by non-zero elements of the ring $\mathbb{Z} \left[ G/G_i \right]$, i.e. $G_i/G_{i+1}$ is a vector space over the skew-field of fractions $Q \left( G/G_i \right)$ of the ring $\mathbb{Z} \left[ G/G_i \right]$ (such a skew-field exists). We will present new results in the model theory of divisible rigid groups.