Irreducible affine varieties over a free group II. Systems in triangular quasi-quadratic form and description of residually free groups

We shall prove the conjecture of Myasnikov and Remeslennikov [4] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group) if and only if it is embeddable in the Lyndon's exponential group F(Z[x]), which is the Z[x]-completion of the free group. Here Z[x] is the ring of polynomials of one variable with integer coefficients. Historically, Lyndon's attempts to solve Tarski's famous problem concerning the elementary equivalence of free groups of different ranks led him to introduce F(Z[x]). An There Exists-free group is a group G such that the class of There Exists-formulas, true in G, is the same as the class of There Exists-formulas, true in a nonabelian free group. A finitely generated group is There Exists-free if and only if it is fully residually free [22]. Our result gives an algebraic description of There Exists-free groups. We shall give an algorithm to represent a solution set of an arbitrary system of equations over F as a union of finite number of irreducible components in the Zariski topology on F(n). The solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms. The possibility of such a decomposition for a solution set was conjectured by Razborov in [20] and also by Rips. We shall give a description of systems of equations determining irreducible components using methods developed in [13, 19]; it is possible to find some of these methods in [18]. We are thankful to E. Rips for attracting our attention to these techniques. (C) 1998 Academic Press.