# On certain C-test words for free groups

Research paper by **Donghi Lee**

Indexed on: **16 Mar '01**Published on: **16 Mar '01**Published in: **Mathematics - Group Theory**

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#### Abstract

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.