Divisible Rigid Groups. IV. Definable Subgroups

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, when 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]. We describe subgroups of a divisible rigid group which are definable in the signature of the theory of groups without parameters and with parameters.