On Certain C-test Words for Free Groups

Abstract. Let Fm be a free group of a finite rank m ≥ 2 and Xi, Yj be elements in Fm. A nonempty word w(x1, . . . , xn) is called a C-test word in n letters for Fm if, whenever w(X1, . . . ,Xn) = w(Y1, . . . , Yn) 6= 1, the two n-tuples (X1, . . . , Xn) and (Y1, . . . , Yn) are conjugate in Fm. In this paper we construct, for each n ≥ 2, a C-test word vn(x1, . . . , xn) with the additional property that vn(X1, . . . , Xn) = 1 if and only if the subgroup of Fm generated by X1, . . . ,Xn is cyclic. Making use of such words vm(x1, . . . , xm) and vm+1(x1, . . . , xm+1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u1, u2 ∈ Fm such that every endomorphism ψ of Fm with non-cyclic image is completely determined by ψ(u1), ψ(u2).