Divisible rigid groups. II. Stability, saturation, and elementary submodels

A group $G$ is said to be rigid if it contains a normal series
$$ G=G_1>G_2>\dots>G_m>G_{m+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and, treated as right $\mathbb Z[G/G_i]$-modules, are torsion-free. A rigid group $G$ is divisible if elements of the quotient $G_i/G_{i+1}$ are divisible by nonzero elements of the ring $\mathbb Z[G/G_i]$. Every rigid group is embedded in a divisible one.
Previously, it was stated that the theory $\mathfrak T_m$ of divisible $m$-rigid groups is complete. Here, it is proved that this theory is $\omega$-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated $m$-rigid groups. Also, it is proved that the theory $\mathfrak T_m$ admits quantifier elimination down to a Boolean combination of $\forall\exists$-formulas.