Fully residually free groups and graphs labeled by infinite words

Let F = F(X) be a free group with basis X and ℤ[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (ℤ[t], X)-graphs - a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich- Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon's group Fℤ[t], the author's representation of elements of F^ℤ[t] by infinite (ℤ[t], X)-words, and Stallings folding method for subgroups of free groups. As an application, we solve the membership problem for finitely generated subgroups of F ^ℤ[t], as well as for finitely generated fully residually free groups.