Divisible rigid groups

Abstract

A soluble group G is rigid if it contains a normal series of the form G = G1 > G2 > � > Gp > Gp+1 = 1, whose quotients Gi/Gi+1 are Abelian and are torsion-free as right Z[G/Gi]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble. In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients Gi/Gi+1 are divisible by any elements of respective groups rings Z[G/Gi]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely up to G-isomorphism.