Divisible rigid groups. Algebraic closedness and elementary theory

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. We prove two theorems.
THEOREM 1. The following three conditions for a group $G$ are equivalent: $G$ is algebraically closed in the class $\Sigma_m$ of all $m$-rigid groups; $G$ is existentially closed in the class $\Sigma_m$; $G$ is a divisible $m$-rigid group.
THEOREM 2. The elementary theory of a class of divisible $m$-rigid groups is complete.