The object of this paper, which is the first in a series of three, is to lay the foundations of the theory of ideals and algebraic sets over groups.

No abstract

The object of this paper, which is the second in a series of three, is to lay the logical foundations of the algebraic geometry over groups. Exploiting links between the algebraic geometry over groups and model theory we solve two problems on geometrical equivalence of groups which are due to B. Plotkin.

In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along the way we lay down foundations of algebraic geometry over arbitrary algebraic structures.

We introduce a family of multiplicative distributions {μs|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μs(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.