Paper detail

On certain C-test words for free groups

Let F_m be a free group of a finite rank m > 1 and X_i, Y_j be elements in F_m. A non-empty word w(x_1,..., x_n) is called a C-test word in n letters for F_m if, whenever w(X_1,..., X_n)=w(Y_1,..., Y_n) not equal to 1, the two n-tuples (X_1,..., X_n) and (Y_1,..., Y_n) are conjugate in F_m. In this paper we construct, for each n > 1, a C-test word v_n(x_1,..., x_n) with the additional property that v_n(X_1,..., X_n)=1 if and only if the subgroup of F_m generated by X_1,..., X_n is cyclic. Making use of such words v_m(x_1,..., x_m) and v_{m+1}(x_1,..., x_{m+1}), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u_1, u_2 in F_m such that every endomorphism of F_m with non-cyclic image is completely determined by its values on u_1, u_2.