On Certain C-Test Words for Free Groups

Let Fm be a free group of a finite rank m ≥ 2 and let Xi, Yj be elements in Fm. A non-empty word w(x1,…,xn) is called a C-test word in n letters for Fm if, whenever (X1,…,Xn) = w(Y1,…,Yn) ≠ 1, the two n-typles (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).