Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.

Abstract. This survey article is supposed to make the reader familiar with the algebraic structure of existentially closed groups in specific group classes, and with the ideas and methods involved in this area of group theory. We shall try to give a fairly complete account of the theory, but there will be a certain emphasis on classes of nilpotent groups, locally finite groups, and extensions. 1.

and Related Areas, sponsored by the Minerva Foundation (Germany)

Abstract. Using algebraic and topological K-theory together with complex C ∗-algebras, we prove that every abelian group may be realized as the centre of a strongly torsion generated group whose integral homology is zero in dimension one and isomorphic to two arbitrarily prescribed abelian groups in dimensions two and three.