Description of Coordinate Groups of Irreducible Algebraic Sets over Free 2-Nilpotent Groups

A convenient pure algebraic description of the coordinate groups of irreducible algebraic sets over a non-Abelian free 2-nilpotent group N of finite rank is given. Note that, in algebraic geometry over an arbitrary group N, it is natural to consider groups containing N as a subgroup (so-called N-groups) and homomorphisms of N-groups which are identical on N (N-homomorphisms). As a corollary, we describe all finitely generated groups H that are universally equivalent to N (with constants from N in the language). Additionally, we give a pure algebraic criterion determining when a finitely generated N-group H that is N-separated by N is, in fact, N-discriminated by N.

The basic concepts of algebraic geometry over groups were described in the works by Baumslag et al. [1], which we follow in the terminology and notation. More general, universal algebraic geometry applicable to arbitrary algebraic systems was initiated by Plotkin, Daniyarova, Myasnikov, and Remeslennikov and has been successfully developed for semigroups, noncommutative rings, semilattices, and graphs. At present, algebraic geometry over groups has become an important research tool in combinatorial, geometric, and model-theoretic group theory. The most thoroughly developed are algebraic-geometric methods for free groups, hyperbolic groups, and partially commutative groups, as well as for metabelian, free solvable, and rigid solvable groups. An open question of fundamental importance in this area is the construction of algebraic geometry over torsion-free nilpotent groups, in particular, over free nilpotent groups. In addition to direct interest in nilpotent groups, the importance of this issue lies in that the Fitting subgroup of a group is often distinguished from by a finite system of equations, so algebraic geometry over the nilpotent group is directly embedded in algebraic geometry over the initial possibly non-nilpotent group . In this paper, we describe the algebraic properties of coordinate groups of algebraic sets and their irreducible components over a group (the Zariski topology over N is Noetherian, so every algebraic set is the finite union of its irreducible components). Due to these results, we can hope that other fundamental problems in algebraic geometry over would be solved, for example, a reasonable description of solution sets of finite systems of equations over would be obtained (despite the Diophantine problem over is unsolvable).
Note that, in algebraic geometry over an arbitrary group , it is natural to consider groups containing as a subgroup (so-called N-groups) and homomorphisms of N-groups that are identical on homomorphisms). Our approach to the description of coordinate groups is based on some discrimination results, which are also of interest in themselves. Recall that an N-group is -separated (N-discriminated) by an N-group if for any , (for any nontrivial ) there exists an N-homomorphism such that ( , ). If N = 1, then we obtain the standard concepts of separation and discrimination, which appear in various areas of group theory: in the theory of group varieties, in combinatorial group theory (groups separated by a class of groups ), in geometric group theory (as limits in Gromov-Hausdorff spaces), etc. In recent years, these concepts have played an important role in the theory of group models (for characterization of groups universally equivalent to a given one) and in algebraic geometry over groups.
In 1967 Baumslag proved that a group H is discriminated by a non-Abelian free group if and only if it is separated by and is commutative transitive, or a CT group (the commutation relation is transitive on the set of nontrivial elements of H, i.e., the centralizers of nontrivial elements of H are Abelian). Later, similar results were found to hold for many other groups (e.g., for torsion-free hyperbolic groups). However, non-Abelian nilpotent groups are never CT groups, since they always have a nontrivial center. Nevertheless, it turned out that the definition of a CT group can be slightly generalized so that the new definition works in classes of nilpotent groups as well. Namely, a group H is called a CT group of level k, , or a group if the centralizer of any element that is not in is Abelian; here, is the th term in the upper central series of H. Specifically, if k = 0, then , so groups are exactly CT groups. Note that, for groups, the centralizers of noncentral elements are Abelian. Clearly, is a group. The concept of a group was introduced in [3]. Proposition 1. Let be a non-Abelian free k-nilpotent group of finite rank. If an -group is -discriminated by , then is a group. In particular, if is a 2-nilpotent group, then an N-group that is N-discriminated by N is a group. The following conjecture, if true, provides an analogue of Baumslag's theorem for 2-nilpotent groups.

Conjecture 1. Let be a non-Abelian free 2-nilpotent group of finite rank and H be a finitely generated N-group. Then H is N-discriminated by if and only if H is a
group that is N-separated by N. Now we describe our main results on algebraic geometry. For this purpose, we need some definitions.
Let be a group from the variety of nilpotent groups of step . The Cartesian power G n = (n copies) is called the affine space over . Let be a set of variables and be the nilpotent product, where is a free nilpotent group in with base X. A sys- tem of equations (or ) over is an arbitrary subset of . An element can be treated as a group word in variables with coefficients from , . An element p = (g 1 , ..., is called a solution of the system S = 1 in if in for every . A subset of the affine space is called a algebraic set over if V is the set of all solutions of a system of equations from ; in this case, we write . Additionally, for , we define Rad(V) = = 1 . Obviously, is always a normal subgroup in . The group is called a coordinate group of the algebraic set V. Note that is a G-group with respect to the natural embedding . Using, as a prebasis of closed sets, all algebraic sets of , we turn into a topological space (Zariski topology). An irreducible algebraic set in is defined in the standard manner. It is well known that, for any linear group , specifically, for every finitely generated nilpotent group, the Zariski topology on is Noetherian; therefore, each algebraic set in is a finite union of irreducible algebraic sets.

Theorem 1.
Let be a non-Abelian free nilpotent group of finite rank and be a finitely generated Ngroup. Then the following conditions are equivalent: (1) is a coordinate group of an algebraic set in for some positive integer k.
is an N-subgroup of a finite direct product of groups for some positive integer k in which N is diagonally embedded.
Finally, we can state the main result. Theorem 2. Let N be a non-Abelian free 2-nilpotent group of finite rank and H be a coordinate group of an irreducible algebraic set from . Then the following conditions are satisfied: (1) H is -discriminated by N.
(2) H is an N-subgroup of a finite direct product of groups for some positive integer in which N is diagonally embedded; moreover, is a -group.
(3) H is an -subgroup of a finite direct product of groups for some positive integer in which N is diagonally embedded; moreover, for any , the intersection is an Abelian normal subgroup in .