Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels

Abstract

A group G is said to be rigid if it contains a normal series G = G₁ > G₂ > . . . > G_m > G_m+1 = 1, whose quotients G_i/G_i+1 are Abelian and, treated as right 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 ℤ[G/G_i]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory ????_m of divisible m-rigid groups is complete. Here, it is proved that this theory is ω-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 ????_m admits quantifier elimination down to a Boolean combination of ∀∃-formulas.