Lyndon's Group is Conjugately Residually Free.

Lyndon's group F[sup Z][x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group F[sup Z][x] is fully residually F; i.e. for any finite collection of nontrivial elements in F[sup Z][x] there exists a homomorphism φ : F[sup Z][x] → F which is the identity on F and maps the given elements of F[sup Z][x] into nontrivial elements of F. The importance of F[sup Z][x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into F[sup Z][x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]). In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map F[sup Z][x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of F[sup Z][x]; moreover, it is closely related to the problem of "lifting solutions" of equations from F to F[sup Z][x], since our result implies that the solutions can indeed be "lifted" from F to F[sup Z][x] for equations of the type x[sup -1] c[sub 1] x = c[sub 2]. The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]). [ABSTRACT FROM AUTHOR]