Irreducible Affine Varieties over a Free Group: I. Irreducibility of Quadratic Equations and Nullstellensatz

Abstract

The object of this paper (which consists of two parts) is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups. We prove that any variety over a free group F can be defined by a finite number of systems of equations S = 1 in triangular form where quadratic words play the role of leading terms. Algebraically, irreducible varieties are exactly the varieties whose coordinate groups are fully residually F . The crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into Lyndon’s free Z[x]-group F. Since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety, and the group F is fully residually F , we obtain a characterization of finitely generated fully residually free groups as subgroups of F. The group F and its subgroups have been studied extensively during the last several years. In particular, every finitely generated subgroup of F (hence, every finitely generated fully residually free group) can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and HNN-extensions of the type, where amalgamated and associated subgroups are free abelian of finite rank. In particular, this implies that every finitely generated fully residually free group is finitely presented. There are three parts to this paper: algebraic geometry over free groups, the theory of free exponential groups, and Makanin-Razborov’s machinery to deal with equations over free groups [11],[14],[13]. The algebraic geometry approach has been shown to be very usefull in dealing with equations over groups. It provides necessary topological means and a method to transcribe geometric notions into pure group-theoretic language. Following Baumslag, Myasnikov, Remeslennikov [1] we use the standard algebraic geometry notions such as variety, Zariski topology, irreducibility of varieties, radicals and coordinate groups. Some of the ideas of the algebraic geometry approach go back to R. Lyndon [9], E. Rips, J. Stallings [16].