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 Lyndonʹs exponential groupFZ[x], which is theZ[x]-completion of the free group. HereZ[x] is the ring of polynomials of one variable with integer coefficients. Historically, Lyndonʹs attempts to solve Tarskiʹs famous problem concerning the elementary equivalence of free groups of different ranks led him to introduceFZ[x]. An -free group is a groupGsuch that the class of -formulas, true inG, is the same as the class of -formulas, true in a nonabelian free group. A finitely generated group is -free if and only if it is fully residually free [[22]]. Our result gives an algebraic description of -free groups. We shall give an algorithm to represent a solution set of an arbitrary system of equations overFas a union of finite number of irreducible components in the Zariski topology onFn. 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.